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)\,,\\\ \,\\\ $ (147)

which is equivalent to the sum of the entanglement entropy of two thermal CFTs
on regions with sizes $L_{1},L_{2}$. The critical time $t_{\rm crt1}$ after
which this becomes the minimal result in eq. (146) is determined by the larger
boundary size $L_{1}$ as

$t_{\rm crt1}=\frac{\beta}{\pi}\text{arccosh}\left(\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\right)\simeq L_{1}-\frac{2\beta}{\pi}\,e^{-2\pi
L_{1}/\beta}\,,$ (148)

where the last approximation is taken in the large temperature, i.e.,
$\beta/L_{1}\to 0$. Beyond this critical time scale, the mutual information
between the left and right defects becomes

$I_{\textrm{\tiny L:R}}^{(\textrm{\tiny H})}\big{|}_{|t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}}|\geq t_{\rm crt1}}=8\log
g-\frac{2c}{3}\log\left(\frac{\beta^{2}}{\pi^{2}\Delta L_{\textrm{\tiny
D}}^{2}}\sinh\left(\frac{\pi L_{1}}{\beta}\right)\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)\,.$ (149)

In the following, let us focus on the situations before this critical time
$t_{\rm crt1}$, e.g., $t_{\textrm{\tiny R}}\sim t_{\textrm{\tiny L}}$. Noting
that we have assumed $L_{1}\geq L_{2}$ in this paper, the minimization
associated with $S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})$ is thus determined by

$\min\left\\{\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\,,\quad\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right\\}\,.$ (150)

This determines another transition time $t_{\rm crt2}$, i.e.,

$t_{\rm crt2}=\frac{\beta}{\pi}\text{arccosh}\left(\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)\,,$ (151)

which is related to the smaller boundary size $L_{2}$. This transition only
happens when $\sinh\left(\frac{\pi L_{2}}{\beta}\right)\geq 1$, e.g., at high
temperatures where $L_{2}/\beta>1$. The time evolution of the holographic
entanglement entropy $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny L, R})}_{a,b})$
is thus summarized as

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})=\begin{cases}S_{2}^{(\textrm{\tiny H})}\,,\quad|t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}}|\leq t_{\rm crt2}\,,\\\ S_{3}^{(\textrm{\tiny
H})}\,,\quad t_{\rm crt2}\leq|t_{\textrm{\tiny R}}-t_{\textrm{\tiny L}}|\leq
t_{\rm crt1}\,,\\\ S_{1}^{(\textrm{\tiny H})}\,,\quad|t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}}|\geq t_{\rm crt1}\,.\\\ \end{cases}$ (152)

Correspondingly, the time evolution of the mutual information between the left
and right defects $I_{\textrm{\tiny L:R}}^{(\textrm{\tiny H})}$ is recast as

$I_{\textrm{\tiny L:R}}^{(\textrm{\tiny H})}=\begin{cases}8\log
g-\frac{4c}{3}\log\left(\frac{\beta}{\pi\Delta L_{\textrm{\tiny
D}}}\,\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny R}}-t_{\textrm{\tiny
L}})\right)\right)\,,\qquad\qquad|t_{\textrm{\tiny R}}-t_{\textrm{\tiny
L}}|\leq t_{\rm crt2}\,,\\\ \,\\\ 8\log
g-\frac{2c}{3}\log\left(\frac{\beta^{2}}{\pi^{2}\Delta L_{\textrm{\tiny
D}}^{2}}\,\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\right)\,,\quad t_{\rm
crt2}\leq|t_{\textrm{\tiny R}}-t_{\textrm{\tiny L}}|\leq t_{\rm crt1}\,,\\\
\,\\\ 8\log g-\frac{2c}{3}\log\left(\frac{\beta^{2}}{\pi^{2}\Delta
L_{\textrm{\tiny D}}^{2}}\,\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)\,,\qquad|t_{\textrm{\tiny R}}-t_{\textrm{\tiny
L}}|\geq t_{\rm crt1}\,.\\\ \end{cases}$ (153)

####  3.2.4 Mutual information in three phases

Figure 20: A summary of all relevant extremal surfaces associated with various
defects in different phases. These surfaces apply for any time in the cold and
warm phases, but only for early times in the hot phase. At late times in the
hot phase, the relevant RT surface consists of four disconnected caps
homologous to each of the regulator intervals, e.g., as appear in
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ for this phase.

Instead of the time evolution, we are more interested in the temperature
dependence. Let us consider the case where the boundary sizes $L_{1},L_{2}$
are fixed and the temperature is increased. As shown by the phase diagram in
figure 6, the bulk dual spacetime passes through two phases transitions from
the cold to warm phase and then from the warm to the hot phase. In order to
see all three phases, the ratio $L_{1}/L_{2}$ cannot be too small. Whereas
there are only the cold and hot phases for smaller values of $L_{1}/L_{2}$,
which is similar to what was shown in the symmetric set-up. The critical ratio
is determined by the value of the triple point of the phase diagram 6. More
explicitly, the approximate value for a generic tension was given in eq. (61).
We note that it reduces to

$\left(\frac{L_{1}}{L_{2}}\right)\Bigg{|}_{\rm triple}=\frac{C_{\textrm{\tiny
CW}}}{C_{\textrm{\tiny WH}}}\simeq\frac{4\pi}{(\pi-
k_{2})^{2}}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)-\frac{k_{1}}{\pi-
k_{2}}+\mathcal{O}\left(\frac{c}{\log g}\right)\,,$ (154)

in the large tension limit. Since we are ignoring the time evolution here, we
choose $t_{\textrm{\tiny R}}=t_{\textrm{\tiny L}}$ to simplify the following
analysis. The mutual information $I_{\textrm{\tiny L:R}}$ between the left and
right sides simply vanishes in the cold and warm phases. In the hot phase, the
holographic entanglement entropy is further simplified as

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})=\min\begin{cases}S_{2}^{(\textrm{\tiny
H})}=\frac{4c}{3}\log\left(\frac{\beta}{\pi\epsilon}\right)\,,\\\
S_{3}^{(\textrm{\tiny
H})}=\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\right)+\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)\,,\\\ \end{cases}$ (155)

where either $S_{1}^{(\textrm{\tiny H})}$ or $S_{4}^{(\textrm{\tiny H})}$
could be dominant because of $\sinh(\pi L_{1}/\beta)\geq\sinh(\pi
C_{\textrm{\tiny CW}})>1$ at the high temperature phase and the transition
between $S_{2}^{(\textrm{\tiny H})}$ and $S_{3}^{(\textrm{\tiny H})}$ is given
by

$T_{\textrm{\tiny
DCFT}}L_{2}=\frac{\text{arcsinh}(1)}{\pi}\simeq\frac{0.8814}{\pi}\equiv
C_{\textrm{\tiny H}}\,.$ (156)

As a result, the mutual information $I_{\textrm{\tiny L:R}}$ with increasing
the boundary temperature is given by

$I_{\textrm{\tiny L:R}}\left(t_{\textrm{\tiny R}}=t_{\textrm{\tiny
L}}\right)=\begin{cases}0\,,\qquad\qquad T_{\textrm{\tiny DCFT}}L_{2}\leq
C_{\textrm{\tiny WH}}\,,\\\ \,\\\ 8\log
g-\frac{4c}{3}\log\left(\frac{\beta}{\pi\Delta L_{\textrm{\tiny
D}}}\,\sqrt{\sinh\left(\frac{\pi L_{2}}{\beta}\right)}\right)\,,\quad
C_{\textrm{\tiny WH}}\leq T_{\textrm{\tiny DCFT}}L_{2}\leq C_{\textrm{\tiny
H}}\,,\\\ \,\\\ 8\log g-\frac{4c}{3}\log\left(\frac{\beta}{\pi\Delta
L_{\textrm{\tiny D}}}\right)\,,\qquad T_{\textrm{\tiny DCFT}}L_{2}\geq
C_{\textrm{\tiny H}}\,.\\\ \end{cases}$ (157)

As we increase the temperature, the correlations between two defects on the
same asymptotic boundary (say $D^{(\textrm{\tiny R})}_{a}$ and
$D^{(\textrm{\tiny R})}_{b}$) decreases. This can be explicitly seen in the
mutual information between them:

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny
R}_{b}}\approx\begin{cases}\frac{2c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)+\frac{2c}{3}\log\left(\frac{\Delta
L_{\textrm{\tiny D}}}{L_{2}}\frac{\pi-k_{2}}{\sqrt{k_{1}\sin(\pi-
k_{2})}}\right)+\mathcal{O}(\left(\frac{L_{2}}{L_{1}}\right)^{2})\,,\quad
T_{\textrm{\tiny DCFT}}L_{2}\leq C_{\textrm{\tiny WH}}\,,\\\ \,\\\ 0\,,\quad
T_{\textrm{\tiny DCFT}}L_{2}\geq C_{\textrm{\tiny WH}}\,.\\\ \end{cases}$
(158)

Here we have used the approximate expressions derived in eqs. (116), and (131)
for the cold and warm phases. As a summary, we list all relevant extremal
surfaces associated with different defects in figure 20.

Finally, it is interesting to ask where is the information about the
braneworlds encoded in the boundary theory. In the cold and warm phases, one
can see from the extremal surfaces and hence the corresponding entanglement
wedges in figure 20 that the branes are encoded in the two defects on either
asymptotic boundary. This is expected of course since the branes connect the
two defects on the same boundary. In the hot phase, the brane geometry changes
so that it connects defects on different boundaries and the corresponding
entanglement wedges now indicate that the braneworlds are encoded in those
same defects, e.g., $D^{(\textrm{\tiny R})}_{a}$ and $D^{(\textrm{\tiny
L})}_{a}$. At late times in the hot phase, the relevant RT surfaces for any
combination of two defects consist of two disconnected caps homologous to each
of the regulator intervals, e.g., see $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})$ in figure 20. At this stage, the bulk of the braneworlds is
encoded in the thermal baths formed by the boundary CFT between the defects –
see Grimaldi:2022suv .

Whereas the brane profile only has three phases (cold, warm and hot) depending
on the dimensionless parameter $T_{{\rm DCFT}}L_{2}$, it is notable that the
entanglement wedge of the four defects $D^{(\textrm{\tiny L,R})}_{a,b}$ has
four relevant phases as shown in figure 20. This comes from the fact that the
transition temperatures of the brane configurations and the RT surfaces are
different. One distinguishing feature of the entanglement wedge transition is
that there is an intermediate phase in which all defects are connected in the
bulk through the entanglement wedge. That is, the four defects combined also
encode a large portion of the bulk between the two branes. This phase is
realized in the regime $C_{\textrm{\tiny WH}}\leq T_{\textrm{\tiny
DCFT}}L_{2}\leq C_{\textrm{\tiny H}}$ in the hot phase. It is only in this
intermediate phase, that the entanglement wedge of the four defects is
connected. If one further increases the temperature the entanglement wedge
splits into two disjoint components, each of which contains one of the branes.

##  4 Relation to the Hayden-Preskill Protocol

Figure 21: Left: Original Hayden-Preskill setup. Initial state consists of the
EPR state on $RA$ (The reference and the diary) and $BE$ (the original black
hole with early Hawking radiation). The decoupling theorem (160) states that
after the scrambling time, information of the diary is encoded in Hawking
radiation $DE$. Right-Top A DCFT realization of the Hayden-Preskill setup.
$D^{(\textrm{\tiny L})}_{a}$ is identified with the reference system $R$ in
the HP setup. In this first case the region corresponding to the remaining
black hole $C$ contains the other defect $D^{(\textrm{\tiny L})}_{b}\in C$.
The dual bulk configuration can either be the warm phase or the hot phase.
Right-Bottom Another DCFT realization. In this case the other defect is in the
subsystem corresponding to the late radiation $D^{(\textrm{\tiny L})}_{b}\in
D$.

The boundary CFT setup can be considered a holographic realization of the
Hayden-Preskill (HP) thought experiment Hayden:2007cs . This experiment
suggests that after the Page time, information thrown into an evaporating
black hole (referred to as the “diary”) immediately comes out through the
emitting Hawking radiation. The setup includes the Hilbert space of the diary
($A$), the black hole ($B$), and the early Hawking radiation ($E$) emitted.
Since the black hole is old, it is (almost) maximally entangled with $E$. Also
it is useful to introduce a “reference” system $R$ which is maximally
entangled with the diary $A$. After the black hole $B$ absorbs the diary $A$,
it emits late Hawking quanta $D$ and becomes $C$ (called the remaining black
hole). Total Hilbert space on the final time slice consists of these
subsystems, $\mathcal{H}_{\rm tot}=\mathcal{H}_{\textrm{\tiny
R}}\otimes\mathcal{H}_{\textrm{\tiny C}}\otimes\mathcal{H}_{\textrm{\tiny
D}}\otimes\mathcal{H}_{\textrm{\tiny E}}$. Denoting $U_{AB}$ the unitary
matrix for this evaporation dynamics, the resulting state is

$|\mathrm{HP}\rangle=(I_{R}\otimes U_{AB}\otimes
I_{E})|\mathrm{EPR}\rangle_{RA}\otimes|\mathrm{EPR}\rangle_{BE}\,,$ (159)

where $|\mathrm{EPR}\rangle$ denotes the maximally entangled state on the
indicated bipartite system, see the left panel of figure 21. In Hayden:2007cs
, it was shown that if the time evolution $U_{AB}$ is approximated by a random
unitary, following decoupling theorem holds,

$\int
dU||\rho_{RC}-\rho_{R}\otimes\rho_{C}||_{1}^{2}\leq\left(\frac{d_{A}}{d_{D}}\right)^{2}\,,$
(160)

where $d_{A}={\rm dim}\left(\mathcal{H}_{A}\right),d_{D}={\rm
dim}\left(\mathcal{H}_{D}\right)$. This inequality shows that if we collect
sufficiently many Hawking quanta just emitted from the black hole
$d_{D}>d_{A}$, then information of the diary is already in late plus early
Hawking radiation $DE$, since there is no correlation between the remaining
black hole $C$ and the reference system $R$ which is maximally entangled with
the diary initially. The decoupling between $R$ and $C$,
$\rho_{RC}=\rho_{R}\otimes\rho_{C}$ is equivalent to vanishing of the mutual
information $I_{R:C}=0$.

Each subsystem has a natural counterpart in our two entangled defect CFT
setup. First, we identify one of the defects in the left boundary with the
reference system $R=D^{(\textrm{\tiny L})}_{a}$. The rest of the left boundary
corresponds to the original black hole $B$ which has absorbed the other defect
for the diary, $A=D^{(\textrm{\tiny L})}_{b}$. Since the black hole
evaporates, we split the region for $B$ into two subsystems, one for the
remaining black hole $C$ and the rest combined with the right boundary is the
Hawking radiation system $DE$. We also assume that the size of the Hilbert
space for $D$ is small compared to that of the remaining black hole, $d_{D}\ll
d_{C}$. There are two possibilities for the choice of $DE$, depending on
whether it contains $D^{(\textrm{\tiny L})}_{b}$ or not, see the right panels
of figure 21. The degrees of freedom in the right panel correspond to early
Hawking radiation. Two defects $D^{(\textrm{\tiny R})}_{a},D^{(\textrm{\tiny
R})}_{b}$ in the right boundary will form Bell pairs with the defects on the
black hole side, which is the key to decoupling. 131313One can try several
other identifications between subsystems in the DCFT setup and the HP. The
first possibility is that D and part of E (say $E_{L}$) are on the left
boundary, and the rest of E is on the right. The undesirable feature of this
identification is that $C$ and $E_{L}$ can communicate directly. The second
possibility is that part of D is on the left and the rest of D and all of E
are on the right. However, in order to keep the entropy of E constant, as in
the HP, the size of the region for E must decrease, which is an undesirable
property. However, we want to emphasize that the detailed match with the HP
setups is not necessary for archiving the information recovery in C on the
left boundary from the degrees of freedom on the right. The purpose of this
section is to confirm the decoupling between the two subregions by computing
the following mutual information between $R$ and $C$, i.e.,

$I_{R:C}=S_{R}+S_{C}-S_{RC}\,,$ (161)

in both the warm and hot phases. In our setup, the mutual information $I_{RC}$
diverges because two subsystems $R$ and $C$ are adjacent, so the ultraviolet
modes are localized near the common boundary of $R$ and $C$. However, these UV
modes have nothing to do with the defects, for example the same divergence
appears in the mutual information for the thermal field double state without
defects. Since we are interested in the correlation between the defect in $R$
and the sub-region $C$ in the current setup, which corresponds to the
correlation between the diary and the black hole in the HP, we extract the
contribution of the defect to the mutual information by subtracting that of
the TFD state,

$\Delta I_{R:C}=I_{R:C}-I^{(0)}_{R:C}\,.$ (162)

Finally, we note that in our setup the Yoshida-Kitaev decoder of Hawking
radiation has a simple realization in the DCFT setup.

###  4.1 On the relation to the original HP setup

While the identification between the two setups is natural, a significant
difference arises: the HP protocol involves a time-dependent process, whereas
our system, consisting of two branes, is static. Nevertheless, we argue that
despite this difference, these two systems share a key property that allows us
to realize HP decoding in our system: the swap of the initial entanglement
shared between the reference system $R$ and the black hole to the entanglement
between $R$ and the degrees of freedom.

In the original Hayden-Preskill setup, the initial entanglement exists between
the reference $R$ and the diary $A$, and between the initial black hole $B$
and the early Hawking radiation $E$, as shown in the figure 21. Shortly after
Alice throws her diary $A$ into the initial black hole $B$, $R$ becomes
entangled with $B$. Subsequently, due to chaotic dynamics, this entanglement
between $R$ and the black hole eventually disappears, as indicated by the
inequality (160). Instead, the reference system becomes entangled with the
late and early Hawking radiation $DE$. This entanglement swapping occurs
because of several factors:

  1. 1.

The chaotic dynamics rapidly evolve the initial state to a typical state
(159). The timescale for this evolution is the scrambling time.

  2. 2.

In a typical state, a subsystem $R$ tends to become maximally entangled with
the largest subsystem which could be either the remaining black hole $C$ or
the radiation $DE$. Since after the Page time $|C|<|DE|$, $R$ must become
entangled with the radiation.

  3. 3.

Once the entanglement between $R$ and $DE$ is established, the correlation
between $R$ and $C$ must vanish due to the monogamy property.

The phenomenon of entanglement swapping manifests itself naturally during the
transition from the warm to the hot phase in our doubly holographic setup. For
example, in the warm phase, two defects located on the same boundary are
connected via the brane in the bulk. This configuration, dictated by the
mapping between two different setups, corresponds to the fact that there is a
strong correlation between $R$ and $A$ before the scrambling dynamics in the
HP setup. This strong correlation is explicitly manifested in the expression
for the mutual information $I_{\mathrm{L}_{a}:\mathrm{L}_{b}}$ (158) between
two defects on the same side. However, as the entanglement between the left
and right systems increases, the transition to the hot phase occurs, leading
to a remarkable entanglement swapping. Specifically, the correlation is now
observed between defects located on different boundaries due to the change in
topology of the brane profile. Remarkably, this pattern mirrors the behavior
observed in the HP protocol.

The DCFT setup also differs slightly from the original HP for the evaporating
black hole in that the entanglement between E on the left boundary and others
on the right boundary keeps increasing in the entanglement temperature. Since
information recovery of a black hole is expected to be a general phenomenon
that always happens as long as the black hole is highly entangled with the
external system, both the HP setup and the DCFT setup are the examples of this
situation. Therefore, one can use the DCFT setup to study aspects of
information recovery of the actual evaporating black hole 141414Note that the
increase in entanglement between C and E in the DCFT setup does not mean that
the DCFT state corresponds to the state of the evaporating black hole before
the page time. This can be understood by noting how the EPR state between the
black hole and the radiation is constructed in the HP setup. Namely, since
after the Page time $|\text{BH}|$ (the dimension of the black hole Hilbert
space) is smaller than that of the Hawking radiation, by choosing the sub-
Hilbert space of HR which is isomorphic to $\mathcal{H}_{\rm BH}$, one can
form an EPR state between them, which is the input state for the HP protocol.
Similarly, in the DCFT setup, the right CFT Hilbert space is the sub-Hilbert
space of the early radiation that is isomorphic to the BH, i.e., the left CFT
Hilbert space. We then form the TFD state between them. In particular, the
density matrix $\rho_{\beta}$ on the right CFT obtained by tracing the left
one is not the maximally mixed state on the radiation Hilbert space, so the
information about the black hole is indeed encoded in the radiation..

###  4.2 The mutual information $I_{R:C}$

####  4.2.1 $I_{R:C}$ in the warm phase

Figure 22: Extremal surfaces in the warm phase when $D^{(\textrm{\tiny
L})}_{b}\in C$. The purple line connecting $R$ and $A$ is the brane in the
bulk. Extremal surfaces are drawn with thick green lines. Left: The extremal
surface for $S_{R}$. Middle The extremal surface for $S_{C}$. Right The
extremal surface for $S_{RC}$.

Let us proceed to calculate the mutual information between $R$ and $C$ during
the warm phase. Given the two possible choices for $D$, depending on whether
$D^{(\textrm{\tiny L})}_{b}\in D$ or $D^{(\textrm{\tiny L})}_{b}\in C$, we
divide our analysis into two distinct cases.

Case I : $D^{(\textrm{\tiny L})}_{b}\in C$: We need to evaluate each component
of the mutual information (161). First, $S_{R}$ is nothing but the defect
entropy, i.e., $S_{R}=S_{{\rm EE}}(D^{(\textrm{\tiny L})}_{a})=S_{\rm defect}$
that is defined in eq. (92). The relevant extremal surfaces for $S_{C},S_{RC}$
are depicted in figure 22. The entropy contributions are determined as
follows:

$S_{C}=S_{{\rm EE}}(D^{(\textrm{\tiny L})}_{a})+S_{{\rm
EE},\beta}(\;\ell\Delta\phi_{1}^{\rm Hor}-|D|),\quad S_{RC}=S_{{\rm
EE},\beta}(\;\ell\Delta\phi_{1}^{\rm Hor}-|D|)\,,$ (163)

where $S_{{\rm EE},\beta}(x)$ denotes a thermal entropy:

$S_{{\rm
EE},\beta}(x)=\frac{c}{3}\log\left[\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi}{\beta}x\right)\right]\,.$
(164)

Here, $|D|$ is the length of the region $D$, and $\Delta\phi_{1}^{\rm Hor}$
denotes the size of the black hole in $\mathcal{S}_{1}$ as defined in eq.
(38). Upon combining the aforementioned expressions, we conclude that the
mutual information $I_{R:C}$ during the warm phase assumes a non-zero value:

$I_{R:C}=2S_{\rm
def}^{(\rm{reg})}\approx\frac{2c}{3}\left(\log\left(\frac{\Delta
L_{\textrm{\tiny
D}}}{\epsilon}\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)+\mathcal{O}(1)\right)\,,$
(165)

and is intricately linked to the dimension of the Hilbert space of the diary
$\mathcal{H}_{A}$. This mutual information is diverging because two subregions
$R$ and $C$ are adjacent. To extract the contribution of the defect we
subtract the same mutual information of the state without defects
$I^{(0)}_{RC}$. Considering the standard BTZ black hole as the holographic
dual of CFT without any defects, the background mutual information
$I_{R:C}^{(0)}$ is given by

$\begin{split}I_{R:C}^{(0)}&\approx 2S_{R}+\mathcal{O}(\Delta L_{\textrm{\tiny
D}})\approx\frac{2c}{3}\log\frac{\Delta L_{D}}{\epsilon}\,.\end{split}$ (166)

The regularized mutual information $\Delta I_{R:C}$ (162) is thus reduced to

$\Delta
I_{R:C}\approx\frac{2c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)=2S_{\rm
def}\,,$ (167)

which implies that degrees of freedom of the defect in $R$ is highly
correlated with $C$. In other words, we can conclude that the decoupling is
not achieved yet in the warm phase.

Figure 23: Extremal surfaces in the warm phase when $D^{(\textrm{\tiny
L})}_{b}\in D$. Left: The extremal surface for $S_{R}$. Middle The extremal
surface for $S_{C}$. Right The extremal surface for $S_{RC}$.

Case II: $D^{(\textrm{\tiny L})}_{b}\in D$: The relevant extremal surfaces are
illustrated in figure 23, where the entropies are expressed as:

$S_{C}=\frac{c}{3}\log\left[\frac{L_{\mathcal{S}_{2}}}{\pi\epsilon}\sin\frac{\pi
L_{2}}{L_{\mathcal{S}_{2}}}\right]+S_{\textrm{\tiny EE},\beta}(L_{1})\,,\quad
S_{RC}=S_{{\rm EE}}(D^{(\textrm{\tiny L})}_{b})+S_{\rm horizon}\,,$ (168)

where the Bekenstein-Hawking entropy in the warm phase $S_{\rm horizon}$ has
been derived in eq. (125). For simplicity, we can choose $|R|=|D|=\Delta
L_{D}$ so that the defect contribution in $S_{RC}$ is identical to $S_{R}$.
The mutual information thus reduces to

$\begin{split}I_{R:C}&=\frac{c}{3}\log\left[\frac{L_{\mathcal{S}_{2}}}{\pi\epsilon}\sin\frac{\pi
L_{2}}{L_{\mathcal{S}_{2}}}\right]+S_{\textrm{\tiny EE},\beta}(L_{1})-S_{\rm
horizon}\,.\end{split}$ (169)

It is straightforward to show that the corresponding contributions from the
BTZ background is given by

${I}_{R:C}^{(0)}\approx\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\right)+\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)-\frac{c}{3}\frac{\pi}{\beta}\left(L_{1}+L_{2}\right)\,.$
(170)

Utilizing previous approximations shown in eqs. (115) and (125), we obtain the
approximate contribution of the defect at the first two orders as follows

$\Delta{I}_{R:C}\approx\frac{c}{3}\log\left(\frac{\sin(\pi-k_{2})}{\pi-
k_{2}}\right)+\frac{c}{3}\frac{\pi(\pi-k_{1}-k_{2})}{\pi-
k_{2}}\frac{L_{2}}{\beta}+\mathcal{O}((L_{2})^{2})\,.$ (171)

From (55) we see that in contrast to the earlier result (165), where
$D^{(\textrm{\tiny L})}_{a}\in C$, the expression (171) remains remarkably
small $\Delta{I}_{R:C}\ll\log g$ in the large tension limit, meaning that most
of the degrees of freedom on the defect $R$ are not correlated with $C$. This
is due to the entanglement between $D^{(\textrm{\tiny L})}_{a}$ and
$D^{(\textrm{\tiny L})}_{b}$, which leads to a strong correlation between $R$
and $D$ rather than $C$. However, it’s important to note that the decoupling
is not fully achieved in the warm phase, as evidenced by the non-vanishing
mutual information $\Delta I_{R:C}$. This is because the mutual information
between two defects $D^{(\textrm{\tiny L})}_{a}$ and $D^{(\textrm{\tiny
L})}_{b}$ is large but not maximal, i.e., $I_{\textrm{\tiny
L}_{a}:\textrm{\tiny L}_{b}}<4\log g$ as manifested in eq. (131), so there is
still room for $R=D^{(\textrm{\tiny L})}_{a}$ and $C$ to correlate. Another
way to argue that information recovery is not possible in this case is to
check the entanglement wedges of the subsystems. From the right panel of
figure 23 we see that the brane ending on the diary is almost contained in the
entanglement wedge of $RC$.

####  4.2.2 The mutual information in the hot phase

On the other hand, the mutual information in hot phase is vanishing so the
Hayden-Preskill decoupling is actually happening.

Case I : $D^{(\textrm{\tiny L})}_{b}\in C$: The extremal surfaces are depicted
in figure 24.

Figure 24: Extremal surfaces in the hot phase when $D^{(\textrm{\tiny
L})}_{b}\in C$. Left: The extremal surface for $S_{R}$. Middle The extremal
surface for $S_{C}$. Right The extremal surface for $S_{RC}$.

The curve for $S_{C}$ shown in Fig 24 intersects with the branes once. To
compute the geodesic length intersecting with a brane, we first fix the
location on the brane and compute the geodesic length on each side, add them
up and extremize the total length. In our setting C consists of two disjoint
pieces $C_{1},C_{2}$, and let $C1$ be the piece containing the defect
$D^{(L)}_{b}$. Let $P_{a}:(\phi,r)=(\phi_{\infty}+\phi_{a},r_{\infty})$ and
$P_{b}:(\phi,r)=(\phi_{\infty}+\phi_{b},r_{\infty})$ be the end points of
$C_{1}$ on the boundary cut off surface $r=r_{\infty}$. Also we pick up a
point
$P_{\sigma}(\phi,r)=(\phi_{\infty}+\Delta\phi_{1}(\sigma),r_{1}(\sigma))$ on
the piece of the brane which ends at the defect $D^{(L)}_{b}$.
$\Delta\phi_{1}(\sigma)$ is

$\Delta\phi_{1}(\sigma)=\int^{\infty}_{\sigma}\phi^{\prime}_{1}(s)ds,$ (172)

and in the hot phase $\Delta\phi_{1}(\sigma)=\Delta\phi_{2}(\sigma)$,
$r_{1}(\sigma)=r_{2}(\sigma)$. The length of the geodesic connecting $P_{a}$
and $P_{\sigma}$ is given by

$D_{a}(\sigma)=\ell\log\left[\frac{r_{\infty}}{R}\right]+\ell\log\left[-\frac{\sqrt{r_{1}^{2}(\sigma)-R^{2}}}{R}+\frac{r_{1}(\sigma)}{R}\cosh\frac{R}{\ell}\left(\Delta\phi_{1}(\sigma)-\phi_{a}\right)\right],$
(173)

see figure 25. The length $D_{b}(\sigma)$ of the geodesic connecting
$P_{\sigma}$ and $P_{b}$ is computed similarly.

The location of the intersection $\sigma=\sigma_{*}$ is determined by
extremizing $D_{{\rm tot}}(\sigma)=D_{a}(\sigma)+D_{b}(\sigma)$. This task
gets simplified in the high temperature limit $\;T(L_{1}+L_{2})\gg 1\;$ where
the the horizon becomes large and the entire curve is getting close to the
horizon. This in particular means $\sigma_{*}\rightarrow 0$. By adding the
contribution of other disjoint geodesic that does not cross the brane, in this
limit the value of the entropy is given by

$S_{C}=\frac{c}{3}\log\frac{\beta}{\pi\epsilon}+\frac{c\pi T_{\textrm{\tiny
BH}}}{3}(L_{1}+L_{2}-|D|)+\frac{c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right).$
(174)

Other entanglement entropies $S_{R}$, $S_{RC}$ can be computed in a similar
manner. The interested mutual information $I_{R:C}$ thus reads

$I_{R:C}=\frac{c}{3}\log\left(\frac{\Delta L_{\textrm{\tiny
D}}\beta}{\pi\;\epsilon^{2}}\right)\,.$ (175)

Therefore the contribution of the defect to the mutual information is
vanishing, i.e., $\Delta I_{R:C}=0$, and we conclude that the decoupling is
achived in the hot phase.

Figure 25: Geodesic in $\mathcal{S}_{1}$ region appearing in the calculation
of $S_{C}$ in the hot phase, when $D^{(\textrm{\tiny L})}_{b}\in C$.The
geodesic ends on the brane at $(\phi,r)=(\phi_{1}(\sigma),r_{1}(\sigma))$. We
have a similar contribution from $\mathcal{S}_{2}$ side.

Case II : $D^{(\textrm{\tiny L})}_{b}\in D$: The extremal surfaces are this
case shown in figure 26. The extremal surfaces are depicted in figure 26. We
have

$S_{R}=S_{{\rm EE}}(D^{(\textrm{\tiny L})}_{a}),\quad
S_{C}=\frac{2c}{3}\log\frac{\beta}{\pi\epsilon}+\frac{c\pi T_{\textrm{\tiny
BH}}}{3}(L_{1}+L_{2}-|D|)$ (176)

and

$S_{RC}=\frac{c}{3}\log\frac{\beta}{\pi\epsilon}+\frac{c\pi T_{\textrm{\tiny
BH}}}{3}(L_{1}+L_{2}-|D|)+\frac{c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.$
(177)

Therefore the mutual information is again given by eq. (175), therefore
$\Delta I_{R:C}$ is again vanishing.

Figure 26: Extremal surfaces in the warm phase when $D^{(\textrm{\tiny
L})}_{b}\in D$. Left: The extremal surface for $S_{R}$. Middle The extremal
surface for $S_{C}$. Right The extremal surface for $S_{RC}$.

###  4.3 Comments on the relation to the Yoshida-Kitaev decorder

In Yoshida:2017non , Yoshida and Kitaev proposed a decoding protocol of
information of the diary $T$ from Hawking radiation $DE$. The ingredients are
in addition to the original system $RCDB$, a copy of the diary $A’$ and the
reference system $R’$. The state on $R’T’$ is again the EPR state. In this
setup, one simulates the dynamics of the black hole on early Hawking radiation
and the copy of the diary to get late radiation $D’$ and the remaining black
hole $C’$ records the outcome $BT’\rightarrow C’D’$. The next step is the
postselection of the total state $|\mathrm{YK}\rangle$ to the EPR state on
$D^{\prime}D$. If this succeeds, the state on $RR^{\prime}$ is also the EPR
state with the probability closed to one, meaning the decoding is successful.
The resulting state is given by

$|\mathrm{YK}\rangle=P_{DD^{\prime}}(I_{R}\otimes U_{AB}\otimes
U^{*}_{A^{\prime}E}\otimes
I_{R^{\prime}})|\mathrm{EPR}\rangle_{RA}|\mathrm{HP}\rangle_{RCDE}\otimes|\mathrm{EPR}\rangle_{R^{\prime}A^{\prime}},$
(178)

where
$P_{DD^{\prime}}=|\mathrm{EPR}\rangle_{DD^{\prime}}\langle\mathrm{EPR}|$. This
map is actually equivalent to the Petz recovery map in sufficiently chaotic
systems Nakayama:2023kgr ; Yoshidaunp .

Figure 27:  Left: Yoshida-Kitaev recovery map for reconstructing the state of
the diary from the Hawking radiation in the Hayden Preskill setup. The
protocol consists of the simulation of the black hole dynamics in
$EA^{\prime}$, and the postselection onto
$|\mathrm{EPR}\rangle_{DD^{\prime}}$. Right: Corresponding DCFT setup. In
particular the postselection in the YK protocol corresponds to separating two
defects on both boundaries $L$ and $R$.

We have shown that depending on the choices of the parameter $(\tau,\gamma)$
and the subregion for the late Hawking radiation $D$, the DCFT state can have
three distinct entanglement structures:

  * •

$|W\rangle_{C}$ : dual to the warm phase when $D^{(\textrm{\tiny L})}_{b}\in
C$. In this case the defect contribution of the mutual information $\Delta
I_{R:C}=4\log g$ is maximal, so the decoupling is not achieved at all.

  * •

$|W\rangle_{D}$ : dual to the warm phase when $D^{(\textrm{\tiny L})}_{b}\in
D$. In this case $I_{R:C}$ is small i.e., , $I_{R:C}\ll 4\log g$ and it
remains finite but non-vanishing. In this sense, the decoupling is almost
archived in this state.

  * •

$|H\rangle_{C}$, $|H\rangle_{D}$: The DCFT state dual to the hot phase when
$D^{(\textrm{\tiny L})}_{b}\in C$ and $D^{(\textrm{\tiny L})}_{b}\in D$
respectively . In this case, the defect contribution to $I_{R:C}$ is
vanishing, and the decoupling is complete in these cases.

When the initial state is $|W\rangle_{C}$, as we increase the entanglement
between two sides $L$ans $R$ it becomes $|H\rangle_{C}$ dual to the hot phase,
therefore the decoupling is archived. We identified this with the Hayden
Preskill decoupling for evaporating black holes. When the initial state is
$|W\rangle_{D}$, an observer collecting all Hawking quanta $DE$ can make it
$|H\rangle_{D}$ by a LOCC, namely separating $R$ and $D$ as well as
$R^{\prime}$ and $D^{\prime}$. In the hot phase, one of the two branes
connects $R=D^{(\textrm{\tiny L})}_{a}$ and $R^{\prime}=D^{(\textrm{\tiny
L})}_{b}$ and the brane hosts a black hole dual to the maximally entangled
state on the defect degrees of freedom $|{\rm EPR}\rangle_{RR^{\prime}}$. The
same is true for $D=D^{(\textrm{\tiny L})}_{a}$ and
$D^{\prime}=D^{(\textrm{\tiny L})}_{b}$. Therefore, $|H\rangle_{D}=|{\rm
EPR}\rangle_{RR^{\prime}}\otimes|{\rm
EPR}\rangle_{DD^{\prime}}\otimes|\psi\rangle_{CC^{\prime}}$ .This is quite
similar to the state $|\mathrm{YK}\rangle$ obtained by applying the recovery
protocol to $|{\rm HP}\rangle$. In particular postselection to the maximally
entangled state on $|\mathrm{EPR}\rangle_{DD^{\prime}}$ has a natural
counterpart in the DCFT setup, namely separating defects $D^{(\textrm{\tiny
L})}_{a}\leftrightarrow D^{(\textrm{\tiny L})}_{b}$ and $D^{(\textrm{\tiny
R})}_{a}\leftrightarrow D^{(\textrm{\tiny R})}_{b}$ because both of these
operations result in $|{\rm EPR}\rangle_{DD^{\prime}}$. This induces quantum
teleportation from $R$ to $R^{\prime}$, implying that the information of the
diary appears in early Hawking radiation (right boundary). The phase
transition of the brane configuration in the bulk is a holographic realization
of the Yoshida-Kitaev decoding protocol.

##  5 Discussion

This paper investigates a system involving two entangled and gravitational
universes using double holography. Our study examines the phase transitions of
brane configurations, particularly the topology transitions of brane geometry.
This type of transition provides a manifestation of the ER=EPR paradigm. A
major focus of our exploration is on the quantum information structure of the
entangled universes, specifically the mutual information between defects in
the boundary theories. The doubly holographic setup is further utilized to
discuss Hayden-Preskill protocols and interpret the Yoshida-Kitaev decoding
protocol.

Section 2 provides a comprehensive review of the holographic construction of
the boundary system, which corresponds to the TFD state of two-dimensional
DCFTs. The section outlines the three bulk phases that the system experiences
as temperature increases. The complete phase diagram is presented in figure 6.
Analytical insights are provided for two specific cases where one of the
lengths of the regions between the defects is significantly larger than the
other, and when the two lengths are equal, in sections 2.3 and 2.4,
respectively. The subsequent section delves into examining the quantum
entanglement between various subsystems within the doubly holographic setup,
quantified through mutual information. In particular, we explore how mutual
information between defects varies throughout various phase transitions. In
the cold and warm phases, we found that the mutual information between defects
on the left and right sides vanishes, while it is non-trivial in the hot
phase, through a careful analysis of possible candidates for RT surfaces. All
relevant RT surfaces are shown in figure 20 with the final analytical results
summarized in section 3.2.4. Section 4 of the paper demonstrates that the
doubly holographic setup can emulate the Hayden-Preskill protocol, with
successful decoupling evidenced by the vanishing mutual information between
the radiation and the remaining black hole. Furthermore, the realization of
the Yoshida-Kitaev decoding protocol is discussed by implementing the
postselection procedure through the adjustment of distances between two
defects in the DCFT setup. This section underscores the potential of the DCFT
setup to simulate and study information recovery protocols in the context of
black hole evaporation. The calculations in the main part of the paper focus
on the bulk perspective. Instead, we explore the brane perspective in Appendix
A. In particular, we investigate the phase transition on the braneworld. By
defining the holographic stress tensor on the brane, we present the modified
relation between the trace of the stress tensor and the Ricci curvature, which
includes additional terms arising from the $T\bar{T}$ term. Notably, we
explicitly show the equivalence of the three descriptions by evaluating the
renormalized action from the brane perspective. Further technical details
appear in appendices B and C. In this last section of the paper, we would like
to list some other intriguing points as well as potential future directions
related to this doubly holographic model.

### Double holography of gravitational collapse

In Appendix A, we investigate the phase transition from the brane perspective.
As the temperature increases, the braneworld geometry undergoes a transition
from an asymptotically AdS2 spacetime, supported in the cold and warm phases,
to a pure AdS2 spacetime in the hot phase, as depicted in figure 32. Notably,
in the cold and warm phases, the brane supports a ’star’ or a localized
cluster of CFT excitations that deform the AdS2 geometry. In contrast, in the
hot phase, the AdS2 braneworld corresponds exactly to the two-sided external
black hole, as is evident from the induced metric in eq. (210). The
temperature increase of the thermal bath naturally corresponds to a process of
thermalization. From the perspective of braneworld gravity, this
thermalization process could be understood as gravitational collapse.

To elaborate, the braneworld supports nontrivial matter contributions in the
cold and warm phases (as discussed in Appendix A.3). As the temperature
increases, the CFT star or the accumulation of CFT excitations on the branes
grows in size and density at the center of the braneworld. Beyond the critical
temperature, this effective star collapses into an AdS2 black hole on the
brane. Although our setup represents a static configuration, it serves as a
doubly holographic framework for studying gravitational collapse and black
hole formation.

Assuming a dynamic process with slowly increasing temperature, the
gravitational collapse and black hole formation on the brane can be understood
as a dual of the boundary thermalization process, i.e., an adiabatic increase
of the boundary temperature. With the warm phase, a black hole is formed in
the bulk $\mathcal{S}_{1}$ region. Then there are two effects as the
temperature increases: the horizon radius grows, i.e., the black hole becomes
larger, and the brane bends closer to the black hole.151515We note that
throughout this process the brane and the black hole remain relatively far
apart, in the large tension limit. In this regime and in the warm phase,
$\sigma_{+}\propto 1/x$ from eq. (285), and $T_{\textrm{\tiny DCFT}}\propto
1/(L_{2}\log(1/x))$ from the discussion around eq. (79). Finally, with the
transition to the hot phase, the brane collapses, falling into the bulk black
hole and creating a low-dimensional black hole on the brane universe.

One of the most interesting lessons or observations is that the bulk
perspective in this double holography provides a bird’s eye view for the black
hole formation. From a higher-dimensional viewpoint, the gravitational
collapse on the braneworld can be interpreted as that the bulk black hole is
attracting the braneworld and pulling it closer. The formation of a black hole
is a result of the collapse of the braneworld into the higher-dimensional
black hole. The low-dimensional black hole on the brane universe is nothing
but the intersection of the higher-dimensional black hole with the lower-
dimensional braneworld, e.g., refer to the rightmost plot in figure 2. The
bulk picture also geometrically illustrates the concept of ER=EPR. When the
black hole is formed, the braneworld serves as a wormhole and establishes a
connection with its entangled counterpart through the interior of the higher-
dimensional black hole in the bulk spacetime. Moreover, this holographic
picture for black hole formation can be applied to any type of spacetime, such
as asymptotically flat or de Sitter spacetime in any dimension. While
technically challenging, it would be very interesting to explore black hole
formation in flat or de Sitter braneworlds.

The braneworlds experience the three distinct phases appearing in the phase
diagram shown in figure 6 if we fix the interval lengths and raise the
temperature of the system. Above, we would like to think of the points of the
phase diagram as denoting distinct points in an adiabatic process of
increasing boundary temperature leading to gravitational collapse on the
brane. An alternative way to emulate the process of black hole formation would
be to fix the temperature and the total boundary length (i.e., $L_{1}+L_{2}$).
Then starting at large $L_{1}/L_{2}$, the system would be in the warm phase
where the brane supports a star, and decreasing $L_{1}/L_{2}$ would eventually
bring us to the hot phase where a black hole appears on the brane. In this
scenario, the defects begin relatively close (i.e., $L_{2}\ll L_{1}$) and they
have moved farther apart as the ratio of lengths is decreased. In the bulk,
the branes (or in particular, the turning points on the branes) move closer
towards the black hole sitting in the middle of the bulk $\mathcal{S}_{1}$
region. Of course, this effect is similar to what happens with an increasing
temperature, as the size of the black hole increases. In any event, this
scenario of moving the defects may give an easier approach to studying
gravitational collapse on the braneworld. A starting point for these studies
may be previous investigations of the two-dimensional worldsheet in the bulk
dual to accelerating Wilson lines in the boundary theory, e.g., see
Jensen:2013ora ; Sonner:2013mba ; Hubeny:2014kma .

### Multi-partite boundary

Figure 28: The holographic dual spacetimes of the defect field theory with
multi-partite boundary. The left and right is the preferred phase at very low
and very high temperature, respectively. Note that the cold phase is a
superposition state of two classical spacetimes. A similar warm phase with a
black hole at the center may also dominate.

A straightforward extension of our setup is to consider a multi-partite
boundary configuration. In such a scenario, one can expect the emergence of
different phases characterized by distinct types of brane profiles. The
simplest case arises in the most symmetric configuration. For instance, we can
consider a defect conformal field theory with a multi-partite boundary
configuration, where each segment of the boundary is of equal length, denoted
by $L_{i}=L_{\text{bdy}}/N$. Figure 28 illustrates an example of this
configuration with $N=6$. Due to the strict symmetry constraints, one might
naively expect the presence of only cold and hot phases. This guess stems from
the expectation that the holographic dual spacetime for a symmetric multi-
partite boundary should reflect the same $Z_{N}$ symmetry, essentially
consisting of $N$ replicas of the cold/hot phase explored in subsection 2.4
for a symmetric boundary with two defects. However, this simplistic
expectation does not always hold. We can find that the warm phase should also
dominate for a sufficiently large $N$.

Even more intriguingly, the bulk spacetime may present the replica symmetry
breaking. To illustrate this feature explicitly, we display the holographic
dual spacetime for very low and very high temperatures in the case of $N=6$ in
figure 28. It’s evident that the two candidates of the classical bulk
spacetime corresponding to the cold phases do not exhibit replica symmetry
$Z_{N}$ akin to that of the multi-partite defect conformal field theory.
However, we propose that instead of considering the holographic bulk dual as a
single spacetime, it should be regarded as a superposition of two distinct
classical spacetimes, as illustrated in figure 28. This superposition state
allows for the retention of replica symmetry of the boundary theory, as
expected for a holographic dual. As the boundary temperature increases, we
anticipate that the superposition state for the warm phase, featuring a bulk
black hole at the center, could also emerge as the dominant saddle point.

### Dynamical branes

Figure 29: The possible preparation of an initial state involving a doubly
holographic spacetime with two black holes may be achieved by gluing two BTZ
bulk spacetimes with different masses along a two-dimensional brane.

The present study is confined to the static configurations, and the
corresponding dual quantum state is described by the TFD state defined in
eq.(1). Consequently, our examination of brane profiles is restricted to
static scenarios. This limitation implies that, in the hot phase, the brane
always falls into the BTZ horizon, as illustrated in figure 5. Essentially,
this setup does not allow for the appearance of a bulk spacetime with two
black holes. This restriction arises from the fact that, the two BTZ black
holes before gluing possess the same temperature, i.e., $m_{1}=m_{2}$, which
leads to a turning point always occurring at the BTZ horizon located at
$r=r_{h}$ (see eq.(46)). This restriction is related to the equilibrium state
of the boundary theory, where the temperature associated with the field theory
in the $L_{1}$ and $L_{2}$ portions is identical.

To overcome this limitation and explore a more general scenario, one can
consider two CFTs with different temperatures and join them at the defects.
Similar setups have been explored, see e.g., Ugajin:2013xxa ;
Erdmenger:2017gdk . It is reasonable to anticipate that the holographic dual
spacetime could be constructed by gluing two BTZs at different temperatures,
as depicted in figure 29. Obviously, this would not be the equilibrium state.
Instead, one can view this as the preparation for an initial state. With the
thermal evolution, we can anticipate that the two bulk black holes with
distinct temperatures should be attracted to each other. Eventually, these two
bulk black holes would converge onto the braneworld in the middle and merge
into a larger black hole, thereby transitioning to the hot phase.

While it would be intriguing to investigate more generic dynamic branes in the
AdS3 bulk spacetime, such as a spinning string in the rotating BTZ black hole
background. There has been some previous investigation of this problem in the
literature, e.g., see Gubser:2002tv ; Kim:2014bga ; Kim:2015bba ;
Bachas:2021tnp ; Maxfield:2022rry ; Papadopoulos:2023kyd . The main challenge
lies in solving the junction conditions for generic time-dependent branes.
Analytical solutions for generic brane profiles become more elusive in dynamic
scenarios. However, symmetric setups can be analytically solved since the
junction conditions reduce to the Neumann boundary condition. To wit,

$K_{ab}^{(1)}=-T_{o}h_{ab}=K_{ab}^{(2)}\,.$ (179)

Specifically, for instance, the solutions of junction conditions in global
AdS3 simplifies to the hypersurfaces parametrized as (see e.g.,
Kawamoto:2023wzj ; Akal:2021foz ; Akal:2020wfl ):

$(A+E)\cos T+D\sin
T+\frac{B\ell}{\sqrt{R^{2}+\ell^{2}}}+\frac{R}{\sqrt{R^{2}+\ell^{2}}}\left(C\sin\Phi+(E-A)\cos\Phi\right)=0\,,$
(180)

where constants $(A,B,C,D,E)$ represent real parameters of the brane profiles.
The sign depends on which side of the brane is the choice of our bulk region.
It is straightforward to check the extrinsic curvature of the hypersurfaces
parametrized by eq. (180) is the following

$K_{ab}=\frac{\pm
B\,h_{ab}}{\sqrt{|B^{2}+C^{2}-D^{2}-4AE|}}\frac{1}{\ell}\,,\,\text{with}\quad
T_{o}\ell=\frac{\pm B}{\sqrt{|B^{2}+C^{2}-D^{2}-4AE|}}\,.$ (181)

To begin investigating generic time-dependent branes, one can explore
solutions for specific backgrounds. For example, considering the case of the
hypersurfaces parametrized by eq.(180), one can observe a rich structure of
brane profiles. Moreover, these branes can be studied in any AdS3 spacetime
since they are locally the same. For instance, the hypersurfaces given by
eq.(180) in the BTZ metric (17) can be expressed as:

$Ae^{-\frac{2r_{h}\phi}{\ell}}r+Be^{-\frac{r_{h}\phi}{\ell}}r_{h}+Er+e^{-\frac{r_{h}\phi}{\ell}}\sqrt{r^{2}-r_{h}^{2}}\left(C\cosh\left(\frac{r_{h}t}{\ell^{2}}\right)-D\sinh\left(\frac{r_{h}t}{\ell^{2}}\right)\right)=0\,,$
(182)

where we only show the results related to the non-rotating BTZ black hole for
simplicity.

Despite the analytical challenges associated with time-dependent branes,
exploring these solutions could offer valuable insights into gravitational
collapse and black hole formation, especially in the context of double
holography. Additionally, the generic dynamic brane solutions could serve as a
starting point for investigating more complex configurations and dynamical
scenarios within the framework of double holography.

### Higher-dimensional Spacetime

The analytical analysis presented so far is limited to the AdS3 bulk spacetime
and a two-dimensional braneworld where gravity is mostly topological (see the
effective gravitational action of the braneworld in Appendix A). The advantage
of this limitation is that we have knowledge of all solutions to the bulk
Einstein equations, i.e., AdS3 spacetime, and can explore all possible phases.
It is natural to explore situations involving higher dimensional spacetime,
although this requires more effort to solve the junction conditions and find
all possible brane profiles. Further, the key technical challenge here is that
the branes will backreact on the bulk geometry in a nontrivial way, i.e.,
beyond three dimensions, the bulk geometry is no longer locally just AdSd+1.
This question has been studied extensively in the context of black holes in
Randall-Sundrum braneworlds Emparan:1999wa ; Emparan:1999fd ; Emparan:2000fn ;
Kanti:2004nr ; Majumdar:2005ba ; Fitzpatrick:2006cd ; Creek:2006je ;
Gregory:2008rf ; Figueras:2011gd  – see also a discussion for the present
context in Grimaldi:2022suv .

However, one interesting observation is that the phase transitions shown in
figure 2 appear to be universal across all dimensions, even though the brane
profiles may become more complicated. One fact about this universality is that
the black hole on the codimension-one braneworld exists only at the
intersection between the braneworld and the bulk black hole, at least for
static black holes. This conclusion is easily proven, as shown below.

Consider a higher dimensional static bulk spacetime with a metric given by

$ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{G(r)}+r^{2}d\Omega^{2}_{d-1}\,,$ (183)

where we can assume that there is a black hole at $r=r_{h}$ with
$F(r_{h})=0=G(r_{h})$. The braneworld as a hypersurface can be parameterized
as

$\Phi=\Phi(r,\phi^{i})\,,$ (184)

where we have assumed a static configuration, i.e., no time dependence, and
have chosen the Gaussian normal coordinate for the spatial directions. It is
obvious to write the induced metric on the brane as

$\begin{split}ds^{2}\big{|}_{\mathcal{B}}&=-F(r)dt^{2}+\frac{dr^{2}}{G(r)}+r^{2}\left(d\Phi^{2}+\gamma_{ij}d\phi^{i}d\phi^{j}\right)\\\
&=-F(r)dt^{2}+\left(\frac{1}{G(r)}+r^{2}\Phi^{\prime}(r)^{2}\right)dr^{2}+r^{2}\gamma_{ij}d\phi^{i}d\phi^{j}\,.\end{split}$
(185)

It is nothing but a d-dimensional spacetime whose metric can be recast as

$ds^{2}\big{|}_{\mathcal{B}}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}\gamma_{ij}d\phi^{i}d\phi^{j}\,.$
(186)

with

$f(r)=F(r)\,,\qquad g(r)=\frac{G(r)}{1+G(r)r^{2}\Phi^{\prime}(r)^{2}}\,.$
(187)

Obviously, there is an event horizon at $r=r_{h}$ with $f(r_{h})=0=g(r_{h})$.
Furthermore, we can see that the brane black hole has the same temperature as
the bulk black hole, since

$T_{\textrm{\tiny
B}}=\frac{\sqrt{f^{\prime}(r)g^{\prime}(r)}}{4\pi}\bigg{|}_{r=r_{h}}=\frac{\sqrt{F^{\prime}(r)G^{\prime}(r)}}{4\pi}\bigg{|}_{r=r_{h}}\equiv
T_{\textrm{\tiny BH}}\,.$ (188)

Here we have assumed that the function $G(r)\Phi^{\prime}(r)^{2}$ vanishes at
$r=r_{h}$, i.e., $\Phi^{\prime}(r)$ is a regular function at $r=r_{h}$.
Otherwise, the divergence $\Phi^{\prime}(r_{h})\to\infty$ indicates the
critical case where the brane (without a black hole) is tangent to the bulk
horizon. We can see that the dimension or even the sign of cosmological
constant does not play any role here. More specific calculations related to
the AdS3 case, which is the main focus of this paper, are also shown in the
Appendix A.

Put the above argument the other way around. The fact is that the Killing
vector $K^{\mu}$ of the static bulk spacetime is also the Killing vector of
the braneworld $k^{\alpha}=h^{\alpha}_{\mu}K^{\mu}$ after projection onto the
brane. Noticing that the surface gravity $\kappa$ is defined by
$K^{\nu}\nabla_{\nu}K^{\mu}=\kappa K^{\mu}$, we can show that the event
horizon on the brane has the same surface gravity due to

$k^{\beta}D_{\beta}k^{\alpha}=\kappa k^{\alpha}\,.$ (189)

In summary, what we have illustrated above is that the event horizon of the
braneworld is (must be) the result of the intersection of the braneworld and
the higher dimensional event horizon in the bulk:

$\text{\bf black hole}=\text{\bf d-dim universe}\cap\text{Black Hole in a
(d+1)-Dim Bulk}$ (190)

This is the holographic interpretation of black hole formation on a
d-dimensional braneworld. Considering the generally dynamic braneworld, the
interesting picture is that the star on a braneworld collapses to a black hole
because the brane universe collides with the black hole in the higher-
dimensional bulk.

### de Sitter or flat branes

In Appendix A, we delve into the braneworld geometries in different phases. A
noteworthy feature is that the braneworld is consistently (asymptotically) AdS
spacetime, as exemplified by eq. (197), where the intrinsic curvature of the
brane geometry is always negative. This characteristic provides a foundation
for the double holography built upon two AdS/CFT correspondences. However, it
is important to acknowledge that this is contingent on the specific choice of
the brane tension satisfying $|T_{o}\ell|<1$. The inclusion of more diverse
braneworld types becomes possible by allowing the brane tension to extend
beyond this range. See e.g., Akal:2020wfl ; Akal:2021dqt ; Kawamoto:2023nki
for some explorations of other types of branes in AdS3.

Taking the symmetric setup in $(d+1)$-dimensional AdS spacetime as an example,
where $K_{ab}\propto h_{ab}$, the Gauss equation of the timelike brane
establishes a connection between its intrinsic curvature and the tension,
namely

$R[h_{ij}]=\frac{d}{d-1}(T_{o})^{2}-\frac{d(d-1)}{\ell^{2}}\,.$ (191)

The nature of the braneworld—whether it is (asymptotically) AdS, flat, or de
Sitter—depends on the regime of the brane tension. Specifically, the
braneworld tends to be (asymptotically) AdS, flat, or de Sitter when
$|T_{o}\ell|<d-1$, $|T_{o}\ell|=d-1$, or $|T_{o}\ell|>d-1$, respectively. This
insight underscores the flexibility in extending the doubly holographic
models. By considering $(d+1)$-dimensional AdS spacetime containing a
$d$-dimensional asymptotically de Sitter or flat braneworld, we open avenues
to explore de Sitter holography or flat holography on the braneworld. Of
course, these setups from the boundary perspective differ from the boundary
CFT or defect CFT. One simple evidence is that the dS brane may not intersect
with the conformal boundary and the corresponding “boundary entropy” defined
in eq. (7) could even be complex Akal:2020wfl ; Akal:2021dqt . Intriguingly,
there is a transition from the AdS/dS braneworld to the flat brane by finely
adjusting the brane’s tension to the critical case. However, achieving
consistent results between the two flat limits of AdS brane and dS brane
remains a puzzle.

###### Acknowledgements.

We are happy to thank Dongsheng Ge, Jonathan Harper, Yan Liu, Akihiro Miyata,
Takato Mori, Tadashi Takayanagi and Zhencheng Wang for useful discussions. We
would also like to thank the organizers and participants of the Qubits on the
Horizon 2 meeting (April 4-8, 2022) for providing the stimulating environment
where this collaboration began. Research at Perimeter Institute is supported
in part by the Government of Canada through the Department of Innovation,
Science and Economic Development Canada and by the Province of Ontario through
the Ministry of Colleges and Universities. RCM is supported in part by a
Discovery Grant from the Natural Sciences and Engineering Research Council of
Canada, and by funding from the BMO Financial Group. RCM and SMR were
supported by the Simons Foundation through the “It from Qubit” collaboration.
SMR and TU are also supported by MEXT-JSPS Grant-in-Aid for Transformative
Research Areas (A) “Extreme Universe”, No. 21H05187. SMR is also supported by
by JSPS KAKENHI Research Activity Start-up Grant Number JP22K20370.

##  Appendix A Aspects of Brane Perspective

Figure 30: Sketch illustrating the Euclidean path integral preparation of
states in cold, warm, and hot phases from the brane perspective.

In this section, we study the three distinct phases from the brane
perspective, where CFTs on the left and right boundaries are coupled to
gravitational theories residing on two distinct two-dimensional branes,
denoted as $\mathcal{B}_{i}$. The Euclidean path integral for preparing the
states from the brane perspective is described by figure 30. In particular,
this figure illustrates the topology change of the brane between these phases.
For example, if we double these geometries to evaluate the Euclidean action,
the brane in the cold and warm phases has the topology of a single tube, while
in the hot phase, the brane forms two separate disks. In particular, the
following analysis serves to illustrate that the braneworld is characterized
by an asymptotically AdS2 geometry during both the cold and warm phases. In
these phases, each braneworld connects the two defects on one of the
asymptotic AdS3 boundaries, such as $D_{a}^{(\textrm{\tiny L})}$ and
$D_{b}^{(\textrm{\tiny L})}$. As discussed below in section A.4, the
deformation of the AdS2 geometry is produced by a ‘star’ formed by CFT
excitations that accumulate at the center of the brane. As the temperature
increases, this star grows in density and shrinks in size. . As the
temperature increases, this star grows in density and shrinks in size.
However, when the bulk spacetime transitions to the hot phase, the braneworld
collapses to form an eternal AdS2 black hole spacetime. This black hole
geometry supports a wormhole bridging two defects on different boundaries, for
example, spanning from $D_{a}^{(\textrm{\tiny L})}$ to $D_{a}^{(\textrm{\tiny
R})}$, as shown in figure 30.

###  A.1 Phase transitions on the braneworld

Beginning with the bulk AdS3 spacetimes (12) with two mass parameters
$m_{1},m_{2}$, the induced metric on the brane jointing two bulk regions
$\mathcal{S}_{1},\mathcal{S}_{2}$ is derived as

$\begin{split}ds^{2}\big{|}_{\mathcal{B}}&=-\sigma
dt^{2}+\frac{4T_{o}^{2}\ell^{4}\,d\sigma^{2}}{16(T_{o}\ell)^{2}(1-(T_{o}\ell)^{2})\sigma^{2}+8(m_{1}+m_{2})(T_{o}\ell)^{2}\sigma-(m_{1}-m_{2})^{2}}\,,\\\
&=-\sigma dt^{2}+\frac{\ell_{\textrm{\tiny
B}}^{2}d\sigma^{2}}{4(\sigma-\sigma_{+})(\sigma-\sigma_{-})}\,,\end{split}$
(192)

where the brane coordinate $\sigma$ is related to the AdS radial direction
through $\sigma=\left(\frac{r_{i}^{2}}{\ell^{2}}-m_{i}\right)$ according to
our gauge choice (24). Additionally, we have defined the brane length scale
$\ell_{\textrm{\tiny B}}$, viz,

$\ell_{\textrm{\tiny B}}=\frac{\ell}{\sqrt{1-(T_{o}\ell)^{2}}}\,.$ (193)

Indeed, the induced metric (192) is singular at $\sigma=\sigma_{\pm}$, in
terms of the coordinates $(t,\sigma)$, where $\sigma_{\pm}$ are defined by eq.
(26), namely

$\\{\sigma_{+},\sigma_{-}\\}=\frac{\ell_{\textrm{\tiny
B}}^{2}}{4\ell^{2}}\left(-(m_{1}+m_{2})\pm\sqrt{\left(\frac{m_{1}-m_{2}}{T_{o}\ell}\right)^{2}+4m_{1}m_{2}}\right)\,.$
(194)

Since $\sigma_{+}\geq 0$ and $\sigma_{-}\leq 0$ when $T_{o}\ell\leq 1$, the
midpoint on the brane at $\sigma=\sigma_{+}$ corresponds to the turning point.
However, it is important to note that this is only a coordinate singularity.
That is, the metric described in terms of either $(t,r)$ or $(t,\sigma)$
coordinate patches only covers half of the brane from the turning point out to
the asymptotic boundary.

It is straightforward to ameliorate this situation by introducing a new radial
coordinate $\rho$ on the brane, defined as

$\rho^{2}=\frac{r_{i}^{2}}{\ell^{2}}-m_{i}-\sigma_{+}=\sigma-\sigma_{+}\,.$
(195)

In terms of this radial coordinate $\rho$, $\rho>0$ corresponds to one half of
the brane profile, while $\rho<0$ represents the other half. The turning point
of the brane profile in the bulk corresponds to the origin located at
$\rho=0$. Importantly, this new coordinate $\rho$ results in a smooth brane
geometry, viz,

$ds^{2}\big{|}_{\mathcal{B}}=-(\rho^{2}+\sigma_{+})dt^{2}+\frac{\ell_{\textrm{\tiny
B}}^{2}}{\rho^{2}+\sigma_{+}-\sigma_{-}}d\rho^{2}\,.$ (196)

This expression reveals that the induced geometry asymptotically approaches
AdS2 with a curvature scale of $\ell_{\textrm{\tiny B}}$. Furthermore, the
metric remains well-behaved at the turning point: there are no coordinate
singularities such as those seen in the $r$ or $\sigma$ coordinates. With this
new radial coordinate $\rho$, the Ricci scalar of the brane geometry can be
expressed as

$\begin{split}R[h_{ij}]&=-\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}-\frac{1}{8}\left(\frac{m_{1}-m_{2}}{T_{o}(r_{i}^{2}-m_{i}\ell^{2})}\right)^{2}=-\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}-\frac{(m_{1}-m_{2})^{2}}{8T_{o}^{2}\ell^{4}(\rho^{2}+\sigma_{+})^{2}}\,\\\
\end{split}$ (197)

which asymptotically approaches the constant $-\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}$ as one moves from the center to the boundary of the brane. That is,
the braneworld geometry approaches that of AdS2 in the asymptotic regions. We
note that the Ricci scalar has a position-dependent profile of width
$\Delta\rho\sim\sqrt{\sigma_{+}}$. This deformation is created by excitations
in the braneworld CFT, as we discuss in section A.4. We illustrate this
behaviour with some numerical plots in figure 31.

Figure 31: Ricci scalar of the induced geometry on the brane with fixing
$T_{o}\ell=\frac{9}{10},L_{1}/L_{2}=30$. The blue, orange and red curves are
examples in the cold, warm and hot phases, respectively. The red curve is
simply a constant with $R=-2/\ell_{\textrm{\tiny B}}^{2}$, which matches our
finding in eq. (204) that the braneworld geometry is precisely that of AdS2.

Let us first continue with the analysis of the brane geometry in the cold and
warm phases, where $\sigma_{+}\neq 0$. The conformal metric of the
corresponding brane geometry can be derived as follows:

$ds^{2}\big{|}_{\mathcal{B}}=\frac{\ell_{\textrm{\tiny
B}}^{2}(\rho^{2}(\theta_{\textrm{\tiny
B}})+\sigma_{+})}{\sigma_{+}-\sigma_{-}}\left(-dt_{\textrm{\tiny
B}}^{2}+d\theta_{\textrm{\tiny B}}^{2}\right)\,,$ (198)

where the conformal coordinates are given by 161616$\mathbf{F}(z|m)$ is
referred to as the incomplete elliptic integral of the first kind and defined
by  $\mathbf{F}(z|m):=\int^{z}_{0}\frac{dt}{\sqrt{1-m\sin^{2}t}}\,,\qquad
0<m<1\,.$ (199) The complete elliptic integral of the first kind is then
denoted by $\mathbf{K}(z)$ with
$\mathbf{K}(m)=\mathbf{F}\left(\frac{\pi}{2}|m\right)$ and
$\mathbf{K}(0)=\frac{\pi}{2}$.

$t_{\textrm{\tiny B}}=\frac{\sqrt{\sigma_{+}-\sigma_{-}}}{\ell_{\textrm{\tiny
B}}}t\,,\quad\theta_{\textrm{\tiny
B}}=-i\,\mathbf{F}\left(i\text{arcsinh}\left(\frac{\rho}{\sqrt{\sigma_{+}}}\right)\bigg{|}\frac{\sigma_{+}}{\sigma_{+}-\sigma_{-}}\right)\,.$
(200)

The brane geometry is readily recognizable as an asymptotically AdS2
spacetime, characterized by a radius $\ell_{\textrm{\tiny B}}$. Taking the
infinity $\rho$ limit, it becomes evident that the conformal boundary is
located at

$\lim\limits_{\rho\to\infty}\theta_{\textrm{\tiny
B}}=\mathbf{K}\left(\frac{\sigma_{-}}{\sigma_{-}-\sigma_{+}}\right)\geq\frac{\pi}{2}\,.$
(201)

with the inequality being saturated when $\sigma_{-}=0$. This observation
implies that the Penrose diagram depicting the brane geometry in the cold/warm
phase forms a wider strip compared to that of AdS2, as visually depicted in
figure 32.

Of particular note is the brane geometry associated with the symmetric
boundary configuration where $L_{1}=L_{2}$ (or $m_{1}=m_{2}$).171717Recall
that in the symmetric configuration, we only have the cold phase (i.e., the
warm phase does not exist) for which $m_{1},m_{2}<0$. In this scenario, the
geometry (198) simplifies to global AdS2 whose Ricci scalar (197) reduces to

$R[h_{ij}]=-\frac{2}{\ell_{\textrm{\tiny B}}^{2}}\,,\qquad\text{when}\qquad
m_{1}=m=m_{2}\,.$ (202)

The brane coordinates $(t_{\textrm{\tiny B}},\theta_{\textrm{\tiny B}})$ in
this case correspond to the global AdS2 coordinates in the conformal gauge. To
elaborate further, the global coordinates are given by

$t_{\textrm{\tiny B}}=\frac{\sqrt{-m}}{\ell}t\,,\quad\theta_{\textrm{\tiny
B}}=\arctan\left(\frac{\rho}{\sqrt{\sigma_{+}}}\right)\,,$ (203)

which are obtained by substituting $\sigma_{+}=-\frac{m}{1-T_{o}^{2}\ell^{2}}$
and $\sigma_{-}=0$ into the coordinate transformations presented in eq. (200).
Consequently, the induced metric on the brane can be simplified to

$ds^{2}\big{|}_{\mathcal{B}}=\frac{\ell_{\textrm{\tiny
B}}^{2}}{\cos^{2}\theta_{\textrm{\tiny B}}}\left(-dt_{\textrm{\tiny
B}}^{2}+d\theta_{\textrm{\tiny B}}^{2}\right)\,,$ (204)

which clearly indicates that the brane geometry coincides with global AdS2.
Notably, the conformal boundary of this geometry is positioned at
$\theta_{\textrm{\tiny B}}=\pm\frac{\pi}{2}$, i.e., the locations of the two
defects on the conformal boundary of AdS3.

Another way to verify this conclusion is by employing the explicit brane
profile in the symmetric configuration, as given by eq. (71). Substituting
this brane profile into the bulk AdS3 metric, we can derive the induced metric
in terms of the $(t,r)$ coordinates as follows:

$ds^{2}\big{|}_{\mathcal{B}}=-\left(\frac{r^{2}}{\ell^{2}}-m\right)dt^{2}+\frac{dr^{2}}{(\frac{r^{2}}{\ell^{2}}-m)-\frac{T_{o}^{2}}{r^{2}}(r^{2}-m\ell^{2})^{2}}\,,$
(205)

whose curvature is a negative constant given by eq. (202). To understand how
the original $(t,r)$ coordinates cover the same region as the global
coordinates $(t_{\textrm{\tiny B}},\theta_{\textrm{\tiny B}})$, we can
explicitly express the coordinate transformations between the two systems as

$\frac{r}{\sqrt{-m}\ell}=\sqrt{\frac{1}{(1-(T_{o}\ell)^{2})\cos^{2}\theta_{\textrm{\tiny
B}}}-1}\geq\sqrt{\frac{(T_{o}\ell)^{2}}{1-(T_{o}\ell)^{2}}}\,,$ (206)

or equivalently,

$\cos\theta_{\textrm{\tiny
B}}=\frac{1}{1-(T_{o}\ell)^{2}}\frac{1}{\sqrt{1-\frac{r^{2}}{m\ell^{2}}}}\in[0,1]\,.$
(207)

Here, it is important to note that the minimal radius is located at the
midpoint of the brane. By comparing these transformations, it becomes evident
that the induced metric (205) is equivalent to eq. (204). This demonstrates
the consistency of the results and supports the conclusion that the brane
geometry related to symmetric boundary corresponds to global AdS2 with the
conformal boundary at $\theta_{\textrm{\tiny B}}=\pm\frac{\pi}{2}$.

In summary, we find that the brane geometry during the cold and warm phases
corresponds to an asymptotically thermal AdS2. Notably, the Euclidean time on
the brane is compactified. Furthermore, in the case of symmetric configuration
with $L_{1}=L_{2}$, the brane geometry simplifies to global AdS2.

Figure 32: Phases transitions on the braneworld.

On the other hand, in the high-temperature regime, the bulk spacetime is
dominated by the hot phase. A key characteristic of this phase is that the
brane extends from the defects on the conformal boundary to the black hole
horizon, as illustrated in figure 5. This feature is explicitly reflected by
the position of the middle point on the brane, i.e., $\sigma=\sigma_{+}=0$,
corresponding to $r=r_{h}$ due to $m_{1}=m_{2}=(2\pi T_{\textrm{\tiny
DCFT}}\ell)^{2}$. Considering the universal formula (197), we can immediately
find that the brane geometry in the hot phase is always AdS2. Furthermore, it
is natural to expect that the brane also contains a two-dimensional black
hole, stemming from the dimensional reduction of the black hole interior of a
three-dimensional BTZ black hole. Consequently, the intersection of the brane
and BTZ black hole serves as the horizon of the two-dimensional brane black
hole.

For a more detailed description, the brane profile during the hot phase, as
given by eq. (75) allows us to derive the induced metric as follows:

$ds^{2}\big{|}_{\mathcal{B}}=-\frac{r^{2}-r_{h}^{2}}{\ell^{2}}dt^{2}+\frac{\ell^{2}dr^{2}}{(r^{2}-r_{h}^{2})-\frac{T_{o}^{2}\ell^{2}}{r^{2}}(r^{2}-r_{h}^{2})^{2}}\,.$
(208)

Here, the location of the black hole horizon is identified as $r=r_{h}$, i.e.,
the horizon of the BTZ black hole. It is evident that the curvature of the
two-dimensional brane during the hot phase reduces to the constant defined in
eq. (202). The periodicity of the Euclidean time on the brane remains
unchanged and matches that of the BTZ geometry. This is owing to the fact the
holographic bulk spacetime in the hot phase corresponds to an equilibrium
state.

The geometry of the brane, as defined by the metric (208), accommodates a
static two-dimensional black hole. The temperature of this black hole
reads181818Indeed, the conclusion that black holes on both the brane and in
the bulk share the same temperature is quite general and robust. This outcome
holds true for any static brane profile that intersects the horizon of the
bulk black hole, e.g., see Chen:2020uac ; Chen:2020hmv . This consistency in
temperatures between the two perspectives is naturally anticipated due to the
nature of static configurations.

$T_{\textrm{\tiny
B}}=\frac{\sqrt{-\partial_{r}h_{tt}\,\partial_{r}h^{rr}}}{4\pi}\bigg{|}_{r=r_{h}}=\frac{r_{h}}{2\pi\ell^{2}}=T_{\textrm{\tiny
BH}}=T_{\textrm{\tiny DCFT}}\,,$ (209)

which is determined by the periodicity of the Euclidean $t$ coordinate.

To further elucidate, the comparison with the induced metric during the cold
phase prompts us to reformulate eq. (208) in the AdS-Rindler form, i.e.,

$ds^{2}\big{|}_{\mathcal{B}}=-\rho^{2}dt^{2}+\frac{\ell_{\textrm{\tiny
B}}^{2}}{\rho^{2}-\sigma_{-}}d\rho^{2}=\ell_{\textrm{\tiny
B}}^{2}\left(\frac{\sqrt{m}}{\sinh(\sqrt{m}x_{\textrm{\tiny B}}\
)}\right)^{2}\left(-dt_{\textrm{\tiny B}}^{2}+dx_{\textrm{\tiny
B}}^{2}\right)\,.$ (210)

The dimensionless coordinates $(t_{\textrm{\tiny B}},x_{\textrm{\tiny B}})$
are defined by

$t_{\textrm{\tiny B}}=\frac{t}{\ell}\,,\quad x_{\textrm{\tiny
B}}=\frac{\text{arccoth}\left(\sqrt{1-\frac{\rho^{2}}{\sigma_{-}}}\right)}{\sqrt{m}}=\frac{\ell}{r_{h}}\text{arccoth}\left(\frac{\sqrt{r_{h}^{2}\ell_{\textrm{\tiny
B}}^{2}+(r^{2}-r_{h}^{2})\ell^{2}}}{r_{h}\ell_{\textrm{\tiny B}}}\right)\,.$
(211)

It is important to note that this coordinate system only covers one-sided
black hole exteriors, as depicted in figure 32. In the global coordinates, the
two-sided AdS2 black hole connects the two defects on the left and right
boundaries. For instance, the constant time slice $t=0$ provides a
visualization from the brane perspective, as illustrated by the path integral
in figure 30.

In conclusion, the braneworld in the hot phase is characterized by an eternal
AdS2 black hole, with a temperature identical to that of the BTZ black hole in
the bulk or DCFT on the boundary. Looking from the brane perspective, the
transition between the cold/warm phase and the hot phase corresponds to the
Hawking-Page transition, where the geometry transforms from an
(asymptotically) thermal AdS2 state to an AdS2 black hole configuration.

We finally remark that the transition on the brane is not the standard
Hawking-Page transition for Einstein gravity in AdSd+1 spacetime. For the
$(d+1)-$dimensional AdS-Schwarzschild black hole, the black hole temperature
is given by $T_{\textrm{\tiny
BH}}=\frac{dr_{h}^{2}+(d-2)\ell^{2}}{4\pi\ell^{2}r_{h}}$ and the corresponding
critical temperature associated with the Hawking-Page transition is shown as

$T_{\textrm{\tiny HP}}=\frac{d-1}{2\pi\ell}\,,\qquad\text{with}\qquad
r_{h}=\ell\,.$ (212)

It is clear that this analysis will not apply to our two-dimensional
braneworlds (with $d=1$). The difference is due to the fact that the
gravitational action on the brane is not given by the standard Einstein-
Hilbert action. To determine the critical temperature, we can evaluate and
compare the free energy for the entire two-dimensional system, i.e., bath plus
gravitational brane universe, for both phases. Of course, we have already
derived the same results using the bulk perspective, i.e., with the three-
dimensional gravity coupled to a brane. In the following subsection A.5, we
instead compute the renormalized action from the brane perspective and show
that it leads to consistent results from the bulk perspective.

###  A.2 Black hole threshold on the braneworld

The AdS2 black hole on the braneworld could only exist when the corresponding
bulk dual spacetime assumes the hot phase. As previously established, the
phase transition between the hot and warm phase is determined by the size of
the small interval as shown in eq. (2.2), i.e.,

$T_{\textrm{\tiny DCFT}}L_{2}\simeq\frac{(\pi-k_{2})\left(\sqrt{(\pi-
k_{2})(\pi-
k_{1}-k_{2})+4\text{arctanh}^{2}(T_{o}\ell)}-2\text{arctanh}(T_{o}\ell)\right)}{(\pi-
k_{1}-k_{2})\pi}\,.$ (213)

Beyond this threshold, the hot phase predominates. Conversely, the hot phase
cannot emerge if the size of the smaller interval, namely $T_{\textrm{\tiny
DCFT}}L_{2}$, falls below the critical size defined above. In those cases, the
braneworld cannot give rise to a black hole. As we approach the fusion limit
with $L_{2}\to 0$, it becomes evident that the Casimir energy associated with
interval $L_{2}$ dominates, and is characterized by

$E_{\rm
Casimir}\simeq-\frac{c}{6}\frac{(\pi-\sqrt{-m_{2}}\,\Delta\phi_{2})^{2}}{\pi
L_{2}}\simeq-\frac{c}{6}\frac{(\pi-k_{2})^{2}}{\pi L_{2}}\,.$ (214)

The resultant attractive Casimir force acts as an impediment, restraining the
boundary intervals from attaining a state of thermal equilibrium with AdS2
black hole on the braneworld.

On the contrary, we remark that the braneworld would only support the black
hole phase when the smaller interval $L_{2}$ is relatively large. More
explicitly, a loose upper bound is given by the transition between the hot and
cold phase at $L_{1}=L_{2}$, i.e.,

$T_{\textrm{\tiny DCFT}}L_{2}\geq T_{\textrm{\tiny
CH}}L_{2}=\frac{1}{2\pi}\left(\sqrt{\pi^{2}+4\,\text{arctanh}^{2}\left(T_{o}\ell\right)}-2\,\text{arctanh}\left(T_{o}\ell\right)\right)\,,$
(215)

where the critical temperature $T_{\textrm{\tiny CH}}$ was derived in eq.
(88). Beyond this dimensionless size $T_{\textrm{\tiny DCFT}}L_{2}$, we always
have a black hole on the two-dimensional braneworld.

###  A.3 Brane stress tensors

The standard procedure for evaluating the stress tensor in holographic CFTs
involves considering a CFT coupled to a fixed background metric, e.g.,

$\int D\phi\ e^{iI(\phi,g^{(0)}_{ab})}=e^{iI_{\rm gen}(g^{(0)}_{ab})}\,.$
(216)

This equation represents the path integral of the CFT degrees of freedom,
resulting in a generating function $I_{\rm gen}(g^{(0)}_{ab})$ for the CFT
stress tensor correlators, as denoted by:

$\langle T^{ab}\rangle=\frac{2}{\sqrt{-g^{(0)}}}\,\frac{\delta I_{\rm
gen}}{\delta g^{(0)}_{ab}}\,.$ (217)

Nevertheless, the generating function is notably non-local with respect to the
background metric $g^{(0)}_{ab}$. Moreover, the result typically exhibits
ultraviolet (UV) divergences, with the stress tensor correlation function
displaying behaviour like $\langle T^{ab}\rangle\sim 1/\delta^{d}$, where
$\delta$ represents a short-distance cutoff. To remedy this issue, appropriate
counterterms, which are local functionals of the metric, are subtracted, i.e.,

$\int D\phi\ e^{i\left[I(\phi,g^{(0)}_{ab})-I_{\rm
ct}(g^{(0)}_{ab})\right]}=e^{i\left[I_{\rm gen}(g^{(0)}_{ab})-I_{\rm
ct}(g^{(0)}_{ab})\right]}=e^{iI^{\prime}_{\rm gen}(g^{(0)}_{ab})}\,.$ (218)

The expectation value of the‘renormalized’ stress tensor is properly given by

$\langle T^{ab}\rangle=\frac{2}{\sqrt{-g^{(0)}}}\,\lim_{\delta\to
0}\frac{\delta I^{\prime}_{\rm gen}}{\delta
g^{(0)}_{ab}}=\frac{2}{\sqrt{-g^{(0)}}}\,\lim_{\delta\to 0}\frac{\delta\
}{\delta g^{(0)}_{ab}}\left[I_{\rm gen}-I_{\rm ct}\right]\,.$ (219)

In the context of a holographic CFT, the path integral over the boundary CFT
degrees of freedom can be replaced with the path integral over the bulk
degrees of freedom, particularly the bulk metric. As indicated by the
aforementioned notation, this path integral maintains the non-normalizable
mode of the metric, i.e., the boundary metric $g^{(0)}_{ab}$, fixed.
Furthermore, following the standard assumption, it is assumed that the latter
path integral is evaluated in a saddle point approximation.

We will now discuss the doubly holographic models as described in Chen:2020uac
; Chen:2020hmv  which involve integrating out the bulk theory on both sides of
the brane to produce two boundary CFTs and a dynamical theory of gravity on
the brane. In terms of the boundary theory, the corresponding path integral
can be expressed as

$\begin{split}\int D\tilde{g}\,\int D\phi_{\textrm{\tiny
L}}\,D\phi_{\textrm{\tiny
R}}e^{i\left[I_{\mathcal{B}}(\tilde{g}_{ab})+\sum_{i}I(\phi_{i},\tilde{g}_{ab})\right]}&=e^{i\left[I_{\mathcal{B}}(\tilde{g}_{ab})+\sum_{i}I_{{\rm{gen}},i}(\tilde{g}_{ab})\right]}\\\
&=\int
D\tilde{g}\,e^{i\left[I_{\mathcal{B}}(\tilde{g}_{ab})+\sum_{i}I_{ct,i}(\tilde{g}_{ab})+\sum_{i}\left(I_{{\rm
gen},i}(\tilde{g}_{ab})-I_{{\rm{ct}},i}(\tilde{g}_{ab})\right)\right]}\\\
&=\int D\tilde{g}\,e^{i\left[I_{\rm
grav}(\tilde{g}_{ab})+\sum_{i}I^{\prime}_{{\rm
gen},i}(\tilde{g}_{ab})\right]}\,,\end{split}$ (220)

Here, $I_{\mathcal{B}}$ denotes the intrinsic action on the brane, which
typically takes the form of the worldvolume term:191919However, we may also
add various DGP terms, as described in Chen:2020uac ; Chen:2020hmv .

$I_{\mathcal{B}}=-\frac{T_{o}}{4\pi G_{\textrm{\tiny N}}}\,\int
d^{d}x\,\sqrt{-\tilde{g}}\,.$ (221)

We note that the expressions are written in terms of the induced metric on the
brane $\tilde{g}_{ab}$, rather than the non-normalizable mode $g^{(0)}_{ab}$
which appears above in the standard holographic discussion. Moreover, we
assume that the brane tension is tuned to be near the critical value, which
yields a small UV cutoff $\delta$ and puts us in a regime where the
gravitational theory on the brane is well approximated by Einstein gravity. Of
course, in contrast to the above discussion, the cutoff $\delta$ remains
finite throughout.

As previously discussed, the ‘divergences’ that arise from integrating out the
boundary CFTs or the bulk metrics on either side of the brane are what lead to
the gravitational action on the brane. Specifically, the gravitational brane
action can be expressed as

$I_{\rm grav}=I_{\mathcal{B}}+\sum_{i}I_{{\rm{ct}},i}\,,$ (222)

where we have separated the contributions from the two CFTs. However, it
should be noted that these UV divergences are all local functionals of the
brane metric $\tilde{g}_{ab}$ and therefore will be identical for both
theories.202020While this conclusion is implied by the fact that the bulk
theory, including the cosmological constant, is the same on both sides of the
brane, if we were to consider cases where the bulk theories on either side of
the brane were different, the nature of the two boundary CFTs would also
change. In such a scenario, while the form of the divergences would remain the
same, the precise coefficients of the individual terms may not be identical.
We have introduced the counterterm action as an accounting device to rearrange
the contributions in an interpretable way. In particular, apart from the
gravitational action (222), we have identified the generating functions

$I^{\prime}_{{\rm{gen}},i}=I_{{\rm{gen}},i}-I_{{\rm{ct}},i}\,.$ (223)

which are identical to the expressions derived in the standard holographic
approach. We can further interpret

$\langle T^{ab}_{(i)}\rangle=\frac{2}{\sqrt{-\tilde{g}}}\,\frac{\delta
I^{\prime}_{{\rm{gen}},i}}{\delta\tilde{g}_{ab}}$ (224)

as the expectation value of the CFT with a finite cutoff. These expectation
values serve as sources in the gravitational equations of motion on the brane.

Let us turn our attention to the situation outlined in the main text,
specifically, the case with $d=2$ and a brane metric $h_{ab}$. In this
scenario, the counterterm action comprises a single term, as expressed in the
following equation:

$I_{{\rm{ct}},i}=\frac{1}{8\pi G_{\textrm{\tiny N}}\,\ell}\int
d^{2}x\,\sqrt{-h}=\frac{1}{16\pi G_{\textrm{\tiny eff}}}\int
d^{2}x\,\sqrt{-h}\,\frac{2}{\ell^{2}}\,,$ (225)

where $G_{\textrm{\tiny N}}$ and $G_{\textrm{\tiny eff}}=G_{\textrm{\tiny
N}}/\ell$ represent the Newton’s constants for the bulk and brane theories
Chen:2020uac ; Chen:2020hmv ; Grimaldi:2022suv .212121The Einstein term on the
brane is topological and hence the definition of $G_{\textrm{\tiny eff}}$ is a
convention, e.g., see Grimaldi:2022suv . The term mentioned combines with the
tension term to produce the effective cosmological constant term

$I_{\rm grav}=\frac{1}{4\pi G_{\textrm{\tiny N}}}\int
d^{2}x\,\sqrt{-\tilde{g}}\left(\frac{1}{\ell}-T_{o}\right)=\frac{1}{16\pi
G_{\textrm{\tiny eff}}}\int d^{2}x\,\sqrt{-h}\,\frac{2}{\ell_{\textrm{\tiny
eff}}^{2}}\,,$ (226)

leading to the gravitational equation of motion viz,

$\begin{split}0&=\frac{1}{8\pi G_{\textrm{\tiny
eff}}}\,\frac{1}{\ell_{\textrm{\tiny eff}}^{2}}\,h_{ab}+\langle
T_{ab}^{(1)}\rangle+\langle T_{ab}^{(2)}\rangle\\\ &=\frac{1}{4\pi
G_{\textrm{\tiny N}}}\left(\frac{1}{\ell}-T_{o}\right)h_{ab}+\langle
T_{ab}^{(1)}\rangle+\langle T_{ab}^{(2)}\rangle\,.\end{split}$ (227)

The contribution of the stress tensors here corresponds to the contribution
coming from finite terms, i.e., the $R\log R$ term and higher powers of the
curvature Chen:2020uac ; Chen:2020hmv ; Grimaldi:2022suv .

On the other hand, the total action on the brane takes a simple form:

$I_{\rm brane}=\frac{1}{8\pi G_{\textrm{\tiny N}}}\int
d^{2}x\,\sqrt{-h}\left(K^{(1)}+K^{(2)}-2T_{o}\right)\,,$ (228)

and the corresponding boundary equation of motion corresponds to the Israel
junction condition

$0=\frac{1}{8\pi G_{\textrm{\tiny
N}}}\left(K_{ab}^{(1)}+K_{ab}^{(2)}-\left(K^{(1)}+K^{(2)}-2T_{o}\right)h_{ab}\right)\,.$
(229)

Comparing eqs. (227) and (229), we can arrive at the following identification:

$\langle T_{ab}^{(i)}\rangle=-\frac{1}{8\pi G_{\textrm{\tiny
N}}}\left(K^{(i)}_{ab}-K^{(i)}h_{ab}+\frac{1}{\ell}h_{ab}\right)\,.$ (230)

as elaborated in Kawamoto:2023wzj . If we take the $\delta\to 0$ limit, the
brane stress tensor (230) reduces to the quasi-local stress tensor
Balasubramanian:1999re  (231) appearing in the standard holographic setup,
viz,

$\langle T^{ab}\rangle=\frac{2}{\sqrt{-h}}\frac{\delta I_{\rm
gen}^{\prime}}{\delta\tilde{g}_{ab}}=-\frac{1}{8\pi G_{\textrm{\tiny
N}}}\left(K^{ab}-K\,\tilde{g}^{ab}\right)+\frac{2}{\sqrt{-h}}\frac{\delta
I_{\rm ct}}{\delta h_{ab}}\,.$ (231)

###  A.4 Brane stress tensors in different phases

Focusing on the doubly holographic models considered in the main text, we
explicitly calculate the corresponding brane stress tensor (230) in different
phases in the following. With using the explicit profiles of the codimension-
one brane, we can also obtain the extrinsic curvature of the brane, e.g.,

$\displaystyle K_{ab}^{(i)}=\begin{pmatrix}K_{tt}&0\\\
0&K_{{r_{i}}{r_{i}}}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\left(T_{o}+\frac{(-)^{i}(m_{2}-m_{1})}{4T_{o}\left(r_{i}^{2}-m_{i}\ell^{2}\right)}\right)h_{tt}&0\\\
0&\left(T_{o}-\frac{(-)^{i}(m_{2}-m_{1})}{4T_{o}\left(r_{i}^{2}-m_{i}\ell^{2}\right)}\right)h_{{r_{i}}{r_{i}}}\end{pmatrix}\,,$
(232)

in $(t,r_{i})$ coordinates. Similarly, we have

$K_{ab}^{(i)}=\begin{pmatrix}K_{tt}&0\\\
0&K_{\rho\rho}\end{pmatrix}=\begin{pmatrix}\left(T_{o}+\frac{(-)^{i}(m_{2}-m_{1})}{4T_{o}\ell^{2}(\rho^{2}+\sigma_{+})}\right)h_{tt}&0\\\
0&\left(T_{o}-\frac{(-)^{i}(m_{2}-m_{1})}{4T_{o}\ell^{2}(\rho^{2}+\sigma_{+})}\right)h_{\rho\rho}\end{pmatrix}$
(233)

in $(t,\rho)$ coordinates. It is easy to see that the brane is a CMC slice
Kawamoto:2023wzj  with

$K^{(i)}\equiv K^{(i)}_{ab}h^{ab}=2T_{o}=\text{constant}\,.$ (234)

However, as we show below, the holographic stress tensor on the brane would
only be proportional to the induced metric, i.e., $K^{(i)}_{ab}\propto h_{ab}$
when we have $m_{1}=m_{2}$. However, we further note that $\Delta
K_{ab}=K^{(1)}_{ab}+K^{(2)}_{ab}=2T_{o}\,h_{ab}$, which we comment on below.

It is straightforward to evaluate the brane stress tensor and one obtains

$\begin{split}T_{ab}^{(i)}=\frac{1}{8\pi G_{\textrm{\tiny N}}}\
\begin{pmatrix}\left(T_{o}-\frac{1}{\ell}-\frac{(-)^{i}(m_{2}-m_{1})}{4T_{o}(\rho^{2}+\sigma_{+})}\right)h_{tt}&0\\\
0&\left(T_{o}-\frac{1}{\ell}+\frac{(-)^{i}(m_{2}-m_{1})}{4T_{o}(\rho^{2}+\sigma_{+})}\right)h_{\rho\rho}\end{pmatrix}\,,\end{split}$
(235)

whose trace is given by

$\left(T^{(i)}\right)^{a}{}_{a}=-\frac{1}{8\pi G_{\textrm{\tiny
N}}}\left(\frac{2}{\ell}-K\right)=\frac{1}{4\pi G_{\textrm{\tiny
N}}}\left(T_{o}-\frac{1}{\ell}\right)\,.$ (236)

In the hot phase or a symmetric configuration, the stress tensor reduces to
simply

$T_{ab}^{(i)}=-\frac{1-T_{o}\ell}{8\pi G_{\textrm{\tiny
N}}\ell}h_{ab}\,,\qquad\text{when}\qquad m_{1}=m_{2}\,,$ (237)

which becomes small in the large tension limit (i.e., with $T_{o}\ell\sim 1$).

In contrast to the standard CFT2 result, the trace of the brane stress tensor
in eq. (236) is not simply proportional to the Ricci curvature of the brane
geometry derived in eq. (197). As shown in Kawamoto:2023wzj , the relation
between $T^{a}{}_{a}$ and $R[h]$ is modified by the $T\bar{T}$ term as
follows222222Our definition of the brane stress tensor (235) corresponds to
choosing a vanishing potential of the Liouville field. More explicitly, here,
we can apply eq.(2.22) in Kawamoto:2023wzj  with taking
$T^{(a)}=\frac{1}{\ell}$ and $\mu^{(a)}=0$.

$\begin{split}\left(T^{(i)}\right)^{a}{}_{a}&=\frac{\ell}{16\pi
G_{\textrm{\tiny N}}}\left(R[h]+(8\pi G_{\textrm{\tiny
N}})^{2}\left(T^{(i)ab}\,T^{(i)}_{\,ab}-\left(T^{(i)a}{}_{a}\right)^{2}\right)\right)\,,\\\
&=\frac{c}{24\pi}\left(R[h]+\left(\frac{12\pi\ell}{c}\right)^{2}\left(T^{(i)ab}\,T^{(i)}_{\,ab}-\left(T^{(i)a}{}_{a}\right)^{2}\right)\right)\,.\end{split}$
(238)

We note that the second term, i.e., the $T\bar{T}$ term is suppressed in the
larger tension limit due to the fact that it is quadratic (with including two
traces).

More importantly, the above constraint equation (238) can be understood as the
Einstein equation of the brane geometry. Of course, it is nothing but the
Hamiltonian constraint (Gauss equation) of the timelike brane in AdS3 bulk
spacetime, namely

$R[h]=(K^{(i)})^{2}-K^{(i)ab}K^{(i)}_{\,ab}-\frac{2}{\ell^{2}}\,.$ (239)

Using the explicit solutions of $T_{ij}$ and $R[h]$ derived before, one can
easily verify the equality shown in eq. (238).

Examining the stress tensors in eq. (235), we note the position dependent
contributions with the profile
$\frac{(m_{2}-m_{1})}{4T_{o}(\rho^{2}+\sigma_{+})}$. These excitations of the
braneworld CFTs constitute the ‘star’ to which we referred in earlier
discussions and through the gravitational equation of motion (238) deform the
brane geometry. This profile has a peak value of
${(m_{2}-m_{1})}/{(4T_{o}\sigma_{+})}$ at $\rho=0$, and a width of
$\Delta\rho\sim\sqrt{\sigma_{+}}$. Hence the parameter $\sigma_{+}$, which
determines the turning point of the brane in the bulk, also determines the key
features of the CFT star in the cold and warm phases. We note from eq. (285)
that this parameter is large in the large tension limit. Fixing the lengths
$L_{1}$ and $L_{2}$ in the cold phase, this profile remains unchanged as the
temperature varies in the cold phase. However, in the warm phase, it is
straightforward to see that $\sigma_{+}$ decreases with increasing temperature
in the warm phase. Hence one finds that the profile changes with a growing
peak and a shrinking width as the temperature is increased in this phase.
Hence, it is natural to think that this behaviour is a precursor to
gravitational collapse, as discussed in section 5.

Before concluding this section, we comment on a remarkable property of the
gravitational equation of motion (238). First, we emphasize that the
braneworld supports two copies of the boundary CFT, each with a UV cutoff and
which weakly interact with each other, e.g., see Chen:2020uac ; Chen:2020hmv .
Of course, these are dual to the bulk geometries on either side of the brane.
In eq. (235), we evaluate the stress tensor associated with each CFT
separately. An outstanding feature is that the position-dependent profile
comes with an overall factor of $(-)^{i}$ where $i=1,2$ indicates whether the
CFT is dual to the bulk region $\mathcal{S}_{1}$ or $\mathcal{S}_{2}$. This is
quite remarkable as since the CFT star discussed above generally corresponds
to a positive energy density in CFT1 and a precisely matching negative energy
density in CFT2.232323The signs correspond to the choice of the interval
lengths $L_{1}>L_{2}$. As a result, the total stress tensor on the brane is

$T_{ab}^{(\rm tot)}=T_{ab}^{(1)}+T_{ab}^{(2)}=-\frac{1-T_{o}\ell}{4\pi
G_{\textrm{\tiny N}}\ell}\ h_{ab}\,.$ (240)

That is, the total stress tensor is simply proportional to the induced metric
on the brane.

The feature of the gravitational equation of motion (238) which we wish to
highlight is that the geometry is determined by $T_{ab}^{(1)}$ or
$T_{ab}^{(2)}$ separately, and not the total stress tensor (240). Further,
even though these individual stress tensors differ (i.e., $T_{ab}^{(1)}\neq
T_{ab}^{(2)}$), they consistently yield the same result for the Ricci scalar.
In fact, the present analysis only provides an explicit example of what was
found as a general feature in Kawamoto:2023wzj . For example, eq. (240) is a
result that will hold in general. Of course, this feature of our two-
dimensional gravity theory on the brane contrasts with the intuition that in
higher dimensions, Einstein gravity couples to the total stress tensor, i.e.,
to all forms of energy and momentum simultaneously.

###  A.5 Renormalized action from the braneworld

The phase diagram is determined by evaluating the Euclidean action of each
phase, which is obtained from the three-dimensional gravitational action given
by

$I_{\rm tot}=I_{\rm{bulk}}+I_{\rm{brane}}+I_{\rm{boundary}}\,.$ (241)

The total action is derived from the bulk perspective in the main text. In
accordance with the principle of double holography, we expect to obtain
consistent results from both the brane perspective and the boundary
perspective. In this appendix, we consider the simplest case, namely the
symmetric setup with equal lengths $L_{1}=L_{2}$, from the brane perspective.
Of course, we would reproduce the total action derived in the bulk
perspective, namely

$\begin{split}I_{\textrm{\tiny E}}^{\rm
cold}&=-\frac{c\pi}{6}\frac{1}{T_{\textrm{\tiny BH}}(L_{1}+L_{2})}\,,\\\
I_{\textrm{\tiny E}}^{\rm hot}&=-\frac{c}{6}\pi T_{\textrm{\tiny
BH}}\left(L_{1}+L_{2}\right)-\frac{2c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,,\\\
\end{split}$ (242)

for the cold and hot phases.

The total (Euclidean) action on the brane perspective, denoted as
$\mathcal{I}_{\textrm{\tiny E}}$, can be decomposed into three distinct
components:

$\mathcal{I}_{\textrm{\tiny E}}=\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
bdy CFT}}+\mathcal{I}_{\textrm{\tiny E}}^{\rm
brane\,gravity}+\mathcal{I}_{\textrm{\tiny E}}^{\rm brane\,CFT}\,,$ (243)

where each component corresponds to the contributions from the two-dimensional
conformal field theory residing on the conformal boundary, the effective
gravity on the brane, and the CFTs existing on the braneworld, respectively.

Let us consider the thermal conformal field theory residing on a circular
domain with circumference $L_{\rm bdy}$ ($=L_{1}+L_{2}$ in the present
context). The partition function of this thermal CFT can be obtained in either
the low-temperature or high-temperature limit. The universal expressions for
the thermal partition function are as follows:

$-\ln Z_{\rm thermal}^{\textrm{\tiny
CFT}}=\begin{cases}-\frac{c\pi}{6}\frac{1}{T_{\textrm{\tiny CFT}}L_{\rm
bdy}}\,,\qquad T_{\textrm{\tiny CFT}}\to 0\,,\\\ \,\\\ -\frac{c\pi}{6}L_{\rm
bdy}T_{\textrm{\tiny CFT}}\,,\qquad T_{\textrm{\tiny
CFT}}\to\infty\,.\end{cases}$ (244)

The two expressions are related to each other by the modular transformation
$T_{\textrm{\tiny CFT}}\leftrightarrow 1/(L_{\rm bdy}^{2}T_{\textrm{\tiny
CFT}})$.

The leading behaviour (at $\mathcal{O}(c)$ order) of the thermal partition
function for CFT with large $c$ is universal, assuming the spectrum of light
states is sparse, which is typically true for holographic theories. In the
symmetric case, the DCFT reduces to the boundary conformal field theory on
each side of the defect. Therefore, the partition function of the boundary CFT
living on two boundary circles in the brane perspective can be identified as
follows:

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny bdy
CFT}}=\begin{cases}-\frac{c\pi}{6}\frac{1}{T_{\textrm{\tiny
DCFT}}(L_{1}+L_{2})}+\mathcal{O}(c^{0})\,,\qquad\text{Cold phase}\,,\\\ \,\\\
-\frac{c\pi}{6}(L_{1}+L_{2})T_{\textrm{\tiny
DCFT}}+\mathcal{O}(c^{0})\,,\quad\text{Hot phase}\,.\\\ \end{cases}$ (245)

On the contrary, the two-dimensional brane gravity theory is coupled to the
boundary CFT and can be segregated into the gravitational and CFT sectors. To
calculate the total induced action on the brane, one needs to conduct the
radial integral of the bulk action. In situations where symmetry prevails, the
total unrenormalized action of the brane is derived as

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
AdS}_{2}}\equiv\mathcal{I}_{\textrm{\tiny E}}^{\rm
brane\,gravity}+\mathcal{I}_{\textrm{\tiny E}}^{\rm
brane\,CFT}=-\frac{\ell}{16\pi G_{\textrm{\tiny
N}}}\int_{\mathcal{B}}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\left(-\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}\right)\sqrt{h}\,d^{2}y\,,$ (246)

where $h$ and $\ell_{B}$ represent the induced metric and AdS radius of the
brane $\mathcal{B}$. Note that we have combined the contributions from two
separate branes here. Intuitively, this expression looks like the standard
Einstein-Hilbert term with $R[h]=-\frac{2}{\ell_{\textrm{\tiny B}}^{2}}$ and
an effective Newton constant

$\frac{1}{G_{\rm eff}}\sim\frac{\ell}{G_{\textrm{\tiny
N}}}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.$ (247)

However, the corresponding equation of motion from Einstein-Hilbert action
with an effective Newton constant would not be consistent after this simple
identification. Since the only intrinsic geometric quantity for a two-
dimensional braneworld is its Ricci scalar, we can generally rewrite the
effective gravitational action on the braneworld as a $f(R)$ gravity. The
total action $\mathcal{I}^{\textrm{\tiny AdS}_{2}}$ could be interpreted as
the second-derivative scalar-tensor theory by introducing an extra scalar
field $\Phi$. In particular, the special case with $d=2$ is nothing but the
two-dimensional dilaton gravity without a kinematic term 242424The kinematic
term can emerge after performing the Weyl transformation
$h_{ij}\to\Omega^{2}h_{ij}$.. The corresponding action of the dilaton gravity
(or classically equivalent scalar-tensor theory) is given by

$\begin{gathered}I_{{\rm brane}}^{st}=\frac{1}{16\pi G_{\rm brane}}\int
d^{1+1}y\sqrt{-h}\left(f(\Phi)+f^{\prime}(\Phi)({R}[h_{ij}]-\Phi)\right)\\\
\quad+\frac{1}{8\pi G_{\rm brane}}\oint
dy\sqrt{-\gamma}Kf^{\prime}(\Phi).\end{gathered}$ (248)

with the Newton constant on the brane chosen as

$\frac{1}{G_{\rm brane}}=\frac{\ell}{G_{\textrm{\tiny N}}}$ (249)

and

$f(\Phi)=\Phi\log\left(\frac{\sqrt{4+2\Phi\ell^{2}}+2}{\sqrt{4+2\Phi\ell^{2}}-2}\right)+\frac{1}{T_{o}\ell}\left(\Phi+\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}\right)+\sum_{n>1}c_{n}\left(\Phi+\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}\right)^{n}\,.$ (250)

The important fact here is that the coefficient in the linear term of $f(R)$
function has to be fixed as

$c_{1}=\frac{1}{T_{o}\ell}\,,$ (251)

which needs to reproduce the consistent answer for the intrinsic curvature of
the brane $R[h_{ij}]$. On the contrary, higher-order terms cannot be fixed
since they do not change either the on-shell action or the solution of field
equations.

For example, we can derive the equation of motion with respect to the scalar
field $\Phi$ as

$\Phi=R[h_{ij}]$ (252)

Varying the metric yields the field equations, i.e.,

$0=f(R)-Rf^{\prime}(R)=\frac{R}{\sqrt{1+\frac{\ell^{2}R}{2}}}+\frac{2}{\ell^{2}}\left(\frac{1}{T_{o}\ell}-T_{o}\ell\right)\,,$
(253)

which is an algebraic equation for $R$ . This algebraic equation has a
negative solution

$R=-\frac{2}{\ell^{2}}+2T_{o}^{2}=-\frac{2}{\ell_{\textrm{\tiny B}}^{2}}\,,$
(254)

as we expected for the braneworld. The coefficient associated with the
boundary term is derived as

$\begin{split}f^{\prime}(\Phi)&=\log\left(\frac{\sqrt{4+2\Phi\ell^{2}}+2}{\sqrt{4+2\Phi\ell^{2}}-2}\right)+\frac{1}{T_{o}\ell}-\sqrt{\frac{2}{2+\Phi\ell^{2}}}=\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,,\end{split}$
(255)

where we imposed the on-shell condition, i.e.,
$\Phi=R=-2\frac{1-(T_{o}\ell)^{2}}{\ell^{2}}$. Finally, let us also note that
the effective $f(R)$ gravity on the brane with 252525To make an explicit
connection to the higher curvature gravity shown in Chen:2020uac ;
Grimaldi:2022suv , we note that the on-shell action (in Lorentzian signature)
on the braneworld with the large tension limit ($T_{o}\ell\to 1$) reduces to
$\begin{split}\mathcal{I}^{\textrm{\tiny AdS}_{2}}&=\frac{\ell}{16\pi
G_{\textrm{\tiny
N}}}\int_{\mathcal{B}}R\log\left(\frac{\sqrt{4+2R\ell^{2}}+2}{\sqrt{4+2R\ell^{2}}-2}\right)\sqrt{-h}d^{2}y\\\
&\simeq\frac{\ell}{16\pi G_{\textrm{\tiny
N}}}\int_{\mathcal{B}}\left(\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}+R-R\log\left(-\frac{R\ell^{2}}{8}\right)+\frac{R^{2}\ell^{2}}{4}+\mathcal{O}(R^{3})\right)\sqrt{-h}d^{2}y\,,\end{split}$
(256) where we perform the series expansion with $R\ell^{2}\ll 1$ to obtain
the second line and the first two terms $\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}+R$ are added by hand because they vanish for on-shell brane
solutions. However, the choice of the first two terms is not unique, which
would depend on higher-order terms. In the previous work Chen:2020uac ;
Grimaldi:2022suv , the cosmological constant is chosen as $\frac{2}{\ell_{\rm
eff}^{2}}=\frac{4}{\ell^{2}}\left(1-T_{o}\ell\right)$, resulting in
$\mathcal{I}^{\textrm{\tiny AdS}_{2}}\simeq\frac{1}{16\pi G_{\rm
brane}}\int_{\mathcal{B}}\left(\frac{2}{\ell^{2}_{\rm
eff}}+R-R\log\left(-\frac{R\ell^{2}}{8}\right)+\frac{R^{2}\ell^{2}}{8}+\mathcal{O}(R^{3})\right)\sqrt{-h}d^{2}y\,.$
(257) At this perturbative level (e.g., $\mathcal{O}(R^{2})$), these results
are equivalent due to  $0=\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}+R=\frac{2}{\ell^{2}_{\rm eff}}+R-\frac{\ell^{2}}{2\ell^{4}_{\rm
eff}}\approx\frac{2}{\ell^{2}_{\rm
eff}}+R-\frac{R^{2}\ell^{2}}{8}+\mathcal{O}(R^{3})\,.$ (258) in the large
tension limit.

$f(R)=R\,\log\left(\frac{\sqrt{4+2R\ell^{2}}+2}{\sqrt{4+2R\ell^{2}}-2}\right)+\frac{1}{T_{o}\ell}\left(R+\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}\right)+\sum_{n>1}c_{n}\left(\ R+\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}\right)^{n}\,,$ (259)

or the equivalent scalar-tensor theory (248) indicates that the Wald entropy,
i.e., the area term appearing in the island formula should be given by

$``\text{Area term}"=\frac{f^{\prime}(R)}{4G_{\rm
brane}}\bigg{|}_{R=-\frac{2}{\ell_{\textrm{\tiny
B}}^{2}}}=\frac{\ell}{4G_{\textrm{\tiny
N}}}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)=\frac{c}{6}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\equiv
S_{\rm def}\,,$ (260)

which is the defect entropy $S_{\rm def}$ (7) of a single defect.

On the other hand, the induced action on the brane, denoted by
$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny AdS}_{2}}$, is evidently
divergent since the brane geometry is simply AdS2. To obtain the renormalized
action, a regularization scheme must be introduced for the braneworld theory.
The regulator occurs naturally due to the bulk cut-off surface positioned at
$r=\frac{\ell^{2}}{\delta}$. Similarly, the two-dimensional braneworld
incorporates a cut-off surface located at $r=\frac{\ell^{2}}{\delta}$ on the
brane $\mathcal{B}$. Consequently, it is essential to consider the boundary
term in eq. (248), defined as

$\begin{split}\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny cut-
off}}&=-\frac{\ell}{8\pi G_{\textrm{\tiny
N}}}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\int_{\rm cut-
off}K\sqrt{\gamma}\,d\tau\,.\\\ \end{split}$ (261)

In the cold phase, we have demonstrated that the brane geometry is exclusively
global AdS2. The induced metric can be represented as

$ds^{2}\big{|}_{\mathcal{B}}=-\left(\frac{r^{2}}{\ell^{2}}-m\right)dt^{2}+\frac{dr^{2}}{(\frac{r^{2}}{\ell^{2}}-m)-\frac{T_{o}^{2}}{r^{2}}(r^{2}-m\ell^{2})^{2}}\,,$
(262)

in which the minimal radius equals $r_{\rm
min}=\sqrt{-m}\ell\sqrt{\frac{(T_{o}\ell)^{2}}{1-(T_{o}\ell)^{2}}}$. It is
simple to perform the radial integral and derive

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
AdS}_{2}}=\frac{\Delta\tau\ell}{4\pi G_{\textrm{\tiny
N}}}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\sqrt{\frac{1}{\delta^{2}}+T_{o}^{2}\left(m-\frac{\ell^{2}}{\delta^{2}}\right)}\,.$
(263)

with utilizing the maximal radius on the brane to be
$r=\frac{\ell^{2}}{\delta}$.262626In fact, the choice of the cutoff surface is
not really important for the following calculations. Of course, the Euclidean
time circle is identified as the inverse temperature $\beta=\Delta\tau$. On
the contrary, it can be observed that the contribution from the bulk is
canceled out by the boundary term on the cut-off surface. By substituting the
extrinsic curvature at the cutoff radius, which is represented as

$K\sqrt{\gamma}=\sqrt{(\frac{r^{2}}{\ell^{2}}-m)-\frac{T_{o}^{2}}{r^{2}}(r^{2}-m\ell^{2})^{2}}\,\partial_{r}\sqrt{\gamma}=\sqrt{\frac{r^{2}}{\ell^{4}}+T_{o}^{2}\left(m-\frac{r^{2}}{\ell^{2}}\right)}\,,$
(264)

we find

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny cut-
off}}=-\frac{\beta\ell}{8\pi G_{\textrm{\tiny
N}}}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\sqrt{\frac{1}{\delta^{2}}+T_{o}^{2}\left(m-\frac{\ell^{2}}{\delta^{2}}\right)}\times
2=-\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny AdS}_{2}}\,,$ (265)

which gives rise to the vanishing renormalized action on the brane. We also
emphasize that it is not necessary for this approach to add any additional
counterterms on the brane since there are no extra divergences after we
account for the boundary term $\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
cut-off}}$ on the cut-off surface.

Once the critical temperature is surpassed, the bulk spacetime transitions to
the hot phase in which the brane geometry is exemplified by an AdS2 eternal
black hole. The corresponding induced metric is derived as follows

$ds^{2}\big{|}_{\mathcal{B}}=-\frac{r^{2}-r_{h}^{2}}{\ell^{2}}dt^{2}+\frac{\ell^{2}dr^{2}}{(r^{2}-r_{h}^{2})-\frac{T_{o}^{2}\ell^{2}}{r^{2}}(r^{2}-r_{h}^{2})^{2}}\,.$
(266)

By integrating the radius from the horizon at $r=r_{h}$ to the cut-off
surface, we can reduce the Euclidean action of the braneworld to:

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
AdS}_{2}}=\frac{\Delta\tau}{8\pi G_{\textrm{\tiny
N}}\ell}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\left(\sqrt{r_{\rm
max}^{2}-T_{o}^{2}\ell^{2}(r_{\rm max}^{2}-r_{h}^{2})}-r_{h}\right)\times
2\,.$ (267)

Using the extrinsic curvature at a constant radius in the AdS2 black hole
background, specifically,

$K\sqrt{\gamma}=\frac{\sqrt{r^{2}-T_{o}^{2}\ell^{2}\left(r^{2}-r_{h}^{2}\right)}}{\ell^{2}}\,,$
(268)

we immediately obtain the contribution of the boundary term:

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny cut-off}}=-\frac{\beta}{4\pi
G_{\textrm{\tiny
N}}\ell}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\sqrt{r_{\rm
max}^{2}-T_{o}^{2}\ell^{2}\left(r^{2}_{\rm max}-r_{h}^{2}\right)}\,.$ (269)

Combining the two individual contributions yields the renormalized
gravitational action on the braneworld in the hot phase, i.e.,

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
AdS}_{2}}+\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny cut-off}}=-\frac{\beta
r_{h}}{4\pi G_{\textrm{\tiny
N}}\ell}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.$ (270)

As a summary, we conclude that the renormalized action from the brane
perspective is expressed as

$\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny bdy
CFT}}+\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny
AdS}_{2}}+\mathcal{I}_{\textrm{\tiny E}}^{\textrm{\tiny cut-
off}}=\begin{cases}-\frac{c\pi}{6}\,\frac{1}{T_{\textrm{\tiny
DCFT}}(L_{1}+L_{2})}\,,\qquad\qquad\text{cold phase}\,,\\\ \,\\\
-\frac{c\pi}{6}\,{T_{\textrm{\tiny
DCFT}}(L_{1}+L_{2})}-\frac{c}{6}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,,\quad\text{hot
phase}\,.\end{cases}$ (271)

At the leading order $\mathcal{O}(c)$, the summation of all contributions from
the brane perspective agrees with the analogous Euclidean actions in eqs. (73)
and (77), which were derived with three-dimensional gravitational integrals
from the bulk perspective. Hence, the transition temperature found here for
the symmetric configuration will agree with that found in eq. (78) with
$T_{\text{DCFT}}(L_{1}+L_{2})=C_{\textrm{\tiny CH}}$.

##  Appendix B Around the corner of the defect

In section 2, we decomposed the gravitational action into various constituent
components: the bulk term, the boundary term, and the brane term. However, we
also noted that there were two additional boundary terms associated with the
branes. The first is called the joint term or Hayward term Hayward:1993my ;
Brill:1994mb , which may arise at the intersection between the chosen cut-off
surface and the brane. The second is a boundary counterterm that is added as
part of the brane action.

###  B.1 Boundary terms on the brane

Figure 33: Left: the bulk spacetime $\mathcal{S}_{1}$ with a naive cut-off
surface located at $r=\frac{\ell}{\epsilon}$. Right: the spacetime geometry
$\mathcal{S}_{1}\cup\mathcal{S}_{2}$ near the defect. The black curve
represents the cut-off surface.

To furnish a concrete example, let us consider the symmetric scenario (with
$m_{1}=m_{2}$) for clarity. Without loss of generality, we can parameterize
the bulk regions $\mathcal{S}_{i}$ employing Euclidean global coordinates, as
follows:

$ds^{2}=\left(1+\frac{R^{2}}{\ell^{2}}\right)d\tau^{2}+\frac{dR^{2}}{1+\frac{R^{2}}{\ell^{2}}}+d\Phi^{2}\,.$
(272)

Here, the AdS2 brane is situated at

$|R\cos\Phi|=\frac{T_{o}\ell^{2}}{\sqrt{1-T_{o}^{2}\ell^{2}}}\,.$ (273)

A commonly adopted albeit simple choice for the cut-off surface is expressed
as

$R_{\rm cut}=\frac{\ell^{2}}{\epsilon}\,.$ (274)

as depicted in figure 33. Given the non-smooth nature of this cut-off surface
around the brane, a kink arises on the brane, depicted by the red dot in
figure 33. Consequently, it becomes imperative to incorporate the (timelike)
joint term, often referred to as the Hayward term Hayward:1993my ;
Brill:1994mb , at the intersection between the cut-off surface and the brane.
One can think of this contribution as arising from $\delta$-function
contribution to the extrinsic curvature, which appears in the boundary action
(11). For each side $\mathcal{S}_{i}$, this term takes the following form:

$I_{\rm joint}(\mathcal{S}_{i})=-\frac{1}{8\pi G_{\textrm{\tiny
N}}}\int\theta\sqrt{\gamma}d\tau\,,$ (275)

where $\gamma$ denotes the induced metric on the corner, with
$\sqrt{\gamma}=\sqrt{1+R_{\rm cut}^{2}/\ell^{2}}$, and $\theta$ represents the
angle between the two timelike hypersurfaces, as illustrated in figure 33. By
using the brane profile, it is straightforward to derive the angle $\theta$,
i.e., 272727The gravitational action would not be additive due to obstacles
arising from timelike joints.

$\theta\equiv\arccos\left(n_{\rm cut}\cdot
n_{\mathcal{B}}\right)=\arccos\left(T_{o}\ell\sqrt{1+\epsilon^{2}/\ell^{2}}\right)\,.$
(276)

Evidently, the joint term $I_{\rm joint}$ is divergent (because of the measure
factor $\sqrt{\gamma}$), necessitating the introduction of a counterterm at
the same joint. The simplest choice for this counterterm is given by

$I_{\rm ct}^{\rm joint}(\mathcal{S}_{i})=+\frac{\theta_{o}}{8\pi
G_{\textrm{\tiny N}}}\int\sqrt{\gamma}\,d\tau\,,$ (277)

with $\theta_{o}=\arccos(T_{o}\ell)$. The sum of these two joint terms yields

$I_{\rm joint}+I_{\rm ct}^{\rm
joint}\approx+\frac{T_{o}}{\sqrt{1-T_{o}^{2}\ell^{2}}}\cdot\frac{\beta\epsilon}{16\pi
G_{\textrm{\tiny N}}}+\mathcal{O}(\epsilon^{3})\to 0\,.$ (278)

As a summary, we observe that there is not necessary to concern ourselves with
the Hayward term at the boundary of the brane since it is canceled by the
corresponding counterterm on the joint upon taking the $\epsilon\to 0$ limit.

###  B.2 Smooth cut-off surface

Figure 34: A smooth cut-off surface with regularizing the small part (at the
order of $\mathcal{O}(\epsilon)$) near the defects.

The previous example, which focused on the symmetric case, does not
demonstrate the cancellation of these boundary terms on the brane in general.
It is important to note that a simple regulator like $r_{i}=\ell^{2}/\epsilon$
does not result in a smooth or continuous cutoff surface. Using eq. (24), it
is straightforward to see that in a nonsymmetric configuration, these cutoff
surfaces in the bulk regions $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ do not
intersect the brane as the same radius, i.e., at different values of the brane
coordinate $\sigma$.

To ensure that the cut-off surfaces from $\mathcal{S}_{1}$ to
$\mathcal{S}_{2}$ are connected, we must align the cut-off surfaces at
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ according to

$\frac{\ell}{\epsilon_{1}}=\frac{r_{1}(\sigma)}{\ell}\bigg{|}_{\rm
cut}=\sqrt{\sigma+m_{1}}\,,\qquad\frac{\ell}{\epsilon_{2}}=\frac{r_{2}(\sigma)}{\ell}\bigg{|}_{\rm
cut}=\sqrt{\sigma+m_{2}}\,.$ (279)

However, we would like there to be a single universal cutoff scale in the
defect conformal field theory. Denoting the latter as $\epsilon$, we can
achieve a smooth cut-off surface (as shown in figure 34) by considering two
slightly different cut-off surfaces as follows

$\frac{1}{\epsilon_{i}}=\frac{1}{\epsilon}+A_{i}+B_{i}\epsilon+\mathcal{O}(\epsilon)\,.$
(280)

The specifics of these parameters depend on the chosen smooth cut-off surface
but do not affect any physical quantities (after renormalization) as
considered in the main text. Moreover, a notable advantage of the smooth cut-
off surface is the guarantee of cancellation between the joint term and its
corresponding counterterm at the intersection between the brane and the cut-
off surface. This holds true due to the following equality

$I_{\rm ct}(\mathcal{S}_{1}\cup\mathcal{S}_{2})=-\frac{\pi}{8\pi
G_{\textrm{\tiny N}}}\int\sqrt{\gamma}\,d\tau=-I^{\rm joint}_{\rm
ct}(\mathcal{S}_{1}\cup\mathcal{S}_{2})\,,$ (281)

for any smooth or connected cut-off surface. As a result, we can effectively
disregard the contributions from these boundary terms on the brane.

Lastly, it is worth noting that the choice of the smooth cut-off surface could
influence the calculations of the renormalized action at the order of
$\mathcal{O}(\epsilon)$. A simple choice to achieve a completely smooth cutoff
surface would involve using an extremal hypersurface in the vicinity of the
brane for regularization, as illustrated in figure 34 for the symmetric case.
Since its trace of the extrinsic curvature vanishes, Gibbons-Hawking-York term
on that portion of the cutoff surface vanishes.

##  Appendix C Thermodynamics in the Large Tension Limit

In this appendix, we examine the thermodynamics of our holographic system of
defect CFTs, however, we focus on the large tension limit, i.e.,
$T_{o}\ell\simeq 1$. From the brane perspective, this limit ensures that
contributions coming from the higher curvature terms in the effective
gravitational action (257) on the branes are suppressed, e.g., see
Grimaldi:2022suv . From the bulk perspective, the brane moves far for the
center of the AdS geometry in this limit, as is evident from the growth of the
turning point $\sigma_{+}$ in eq. (26) – see also eq. (285) below. From the
boundary perspective, this limit corresponds to a large number of degrees of
freedom on the defect. That is, the boundary entropy is very large,

$\log g=\frac{c}{12}\,\log\left(\frac{1+T_{o}\ell}{1-T_{o}\ell}\right)\,.$
(282)

Further, with the ratio of $\log g/c\gg 1$, one finds that information leaks
slowly from the defect to the bath in this regime Sully:2020pza .

Introducing the parameter $x=1-T_{o}\ell\ll 1$, we can re-express various
results in our previous analysis using this parameter. For example, the angle
between the boundary defect and the midpoint of the brane in eq. (34) becomes

$\Delta\phi_{i}=\frac{\left[4(1-x)^{2}\,\mathbf{\Pi}\left(-\frac{m_{i}}{\sigma_{+}},\frac{\sigma_{-}}{\sigma_{+}}\right)+\frac{m_{i}-m_{j}}{m_{i}}\left(\mathbf{K}\left(\frac{\sigma_{-}}{\sigma_{+}}\right)-\mathbf{\Pi}\left(-\frac{m_{i}}{\sigma_{+}},\frac{\sigma_{-}}{\sigma_{+}}\right)\right)\right]}{4(1-x)\sqrt{2-x}\,\sqrt{\vphantom{2}x\,\sigma_{+}}}\,.$
(283)

Similarly, $\sigma_{\pm}$ defined in eq. (26) become

$\sigma_{\pm}=\frac{1}{4(2-x)x}\left(-(m_{1}+m_{2})\pm\sqrt{\left(\frac{m_{1}-m_{2}}{1-x}\right)^{2}+4m_{1}m_{2}}\right)\,.$
(284)

In preparing for a more detailed analysis of the various phases, let us
examine the cold/warm phase where $m_{1}+m_{2}<0$. Considering $x\ll 1$, eq.
(284) leads to the following approximations:

$\begin{split}\sigma_{+}&\simeq\frac{|m_{1}+m_{2}|}{4\,x}+\frac{m_{1}^{2}+m_{2}^{2}}{4\,|m_{1}+m_{2}|}+\mathcal{O}(x)\,,\\\
\sigma_{-}&\simeq\frac{(m_{1}-m_{2})^{2}}{8\,|m_{1}+m_{2}|}+x\,\frac{(m_{1}-m_{2})^{2}}{(m_{1}+m_{2})^{2}}\,(3m_{1}+m_{2})(m_{1}+3m_{2})+\mathcal{O}(x^{2})\,.\end{split}$
(285)

Note that $\sigma_{+}\propto 1/x\gg 1$, indicating the brane’s turning point
occurs at very large $\sigma$. Furthermore, in this regime,
$\sigma_{-}/\sigma_{+},\ m_{i}/\sigma_{+}\propto x\ll 1$. Now turning to eq.
(283), we may utilize the generalized power series expansion of the complete
elliptic integral of the third kind, i.e.,

$\mathbf{\Pi}(n,m)=\frac{\pi}{2}\sum^{\infty}_{k=0}\sum_{j=0}^{k}\frac{(2k!)(2j!)m^{j}n^{k-j}}{4^{k}4^{j}(k!)^{2}(j!)^{2}}\,,\qquad|n|<1,\
|m|<1\,.$ (286)

In the limit as $n,m\to 0$, this series expansion becomes

$\mathbf{\Pi}(n,m)\simeq\frac{\pi}{2}\left(1+\frac{m}{4}+\frac{9}{64}m^{2}+\frac{n}{2}+\frac{3}{8}n^{2}+\frac{3}{16}mn\right)+\mathcal{O}(n^{3},m^{3},m^{2}n,n^{2}m)\,.$
(287)

Applying this series expansion to eq. (283) then yields

$\begin{split}\Delta\phi_{1}\simeq\frac{\pi}{\sqrt{2}}\left(\frac{1}{\sqrt{|m_{1}+m_{2}|}}+\frac{(m_{1}-m_{2})(m_{1}+7m_{2})}{8|m_{1}+m_{2}|^{5/2}}x\right)+\mathcal{O}(x^{2})\,,\\\
\Delta\phi_{2}\simeq\frac{\pi}{\sqrt{2}}\left(\frac{1}{\sqrt{|m_{1}+m_{2}|}}+\frac{(m_{2}-m_{1})(7m_{1}+m_{2})}{8|m_{1}+m_{2}|^{5/2}}x\right)+\mathcal{O}(x^{2})\,.\\\
\end{split}$ (288)

for the cold/warm phase with $m_{1}+m_{2}<0$.

Let us note here that in the warm phase, the choice with $m_{1}>0,m_{2}<0$ but
$m_{1}+m_{2}>0$ is unphysical. In this parameter regime and in the large
tension limit with $x\ll 1$, the expansions of $\sigma_{\pm}$ in eq. (284)
interchange their character so that $\sigma_{-}\sim{\mathcal{O}}(1/x)$ and
$\sigma_{+}\sim{\mathcal{O}}(1)$. As a result, we would find that $L_{2}<0$
since

$\Delta\phi_{2}\simeq-\frac{\log x}{\sqrt{2}\sqrt{m_{1}+m_{2}}}\to+\infty\,,$
(289)

using
$\mathbf{K}(-\frac{1}{x})\sim\frac{\sqrt{x}}{2}\log(\frac{16}{x})\sim\mathbf{\Pi}(n,-\frac{1}{x})$
in the limit $x\to 0$.

On the other hand, let us consider $m_{1}>0,m_{2}>0$ which leads to different
approximations:

$\begin{split}\sigma_{+}&\simeq+\frac{(m_{1}-m_{2})^{2}}{8(m_{1}+m_{2})}+x\,\frac{(m_{1}-m_{2})^{2}}{(m_{1}+m_{2})^{2}}\,(3m_{1}+m_{2})(m_{1}+3m_{2})+\mathcal{O}(x^{2})\,,\\\
\sigma_{-}&\simeq-\frac{m_{1}+m_{2}}{4\,x}-\frac{m_{1}^{2}+m_{2}^{2}}{4(m_{1}+m_{2})}+\mathcal{O}(x^{2})\,,\\\
\end{split}$ (290)

indicating the brane’s turning point is located at the middle of bulk
spacetime. Consequently, the expression (283) for $\Delta\phi_{i}$ with
elliptic integrals yields

$\Delta\phi_{i}\simeq\frac{\log\left(\frac{32}{x}\right)+2\log\left(\frac{m_{1}+m_{2}}{|m_{1}-m_{2}|}\right)}{\sqrt{2(m_{1}+m_{2})}}-\frac{1}{\sqrt{m_{i}}}\text{arcsinh}\left(\frac{\sqrt{8m_{i}(m_{1}+m_{2})}}{|m_{1}-m_{2}|}\right)+\mathcal{O}(x^{2})\,.$
(291)

Of course, we are only interested in the case with $m_{1}=m_{2}>0$ for the hot
phase. Then taking the limit $m_{2}\to m_{1}$ above yields

$\Delta\phi_{i}\simeq\frac{\log\left(\frac{2}{x}\right)}{2\sqrt{m_{i}}}+\mathcal{O}(x^{2})\,,$
(292)

This same result can also be derived by taking the large tension limit of the
precise expression in eq. (47).

Cold phase ($m_{i}<0$): The renormalized Euclidean action was given in eq.
(35), as follows:

$I^{\textrm{\tiny cold}}_{\textrm{\tiny E}}=\frac{c}{24\pi T_{\textrm{\tiny
DCFT}}}\left(L_{1}\,m_{1}+L_{2}\,m_{2}\right)\,\frac{1}{\ell^{2}}\,.$ (293)

The parameters $m_{i}$ (or rather $m_{i}/\ell^{2}$) may be determined in terms
of boundary data using eq. (33), i.e.,

$L_{i}=2\ell\left(\frac{\pi}{\sqrt{-m_{i}}}-\Delta\phi_{i}\right)\,.$ (294)

We are considering the $\Delta\phi_{i}$ given by eq. (288) in the limit $x\ll
1$. To leading order, we then find

$L_{1}\simeq
2\pi\ell\left(\frac{1}{\sqrt{|m_{1}|}}-\frac{1}{\sqrt{2\,|m_{1}+m_{2}|}}\right)\,,\quad
L_{2}\simeq
2\pi\ell\left(\frac{1}{\sqrt{|m_{2}|}}-\frac{1}{\sqrt{2\,|m_{1}+m_{2}|}}\right)\,.$
(295)

The solutions $m_{i}(L_{1},L_{2})$ are analytical but complex functions and
hence we refrain from explicitly presenting the general expressions here.
Utilizing standard thermodynamic relations, we derive the energy and entropy
as

$\begin{split}E^{\rm cold}&=-T_{\textrm{\tiny DCFT}}^{2}\frac{\partial
I^{\textrm{\tiny cold}}_{\textrm{\tiny E}}}{\partial T_{\textrm{\tiny
DCFT}}}=\frac{c}{24\pi}\left(L_{1}\,m_{1}+L_{2}\,m_{2}\right)\,\frac{1}{\ell^{2}}\,,\\\
S^{\rm cold}&=-\frac{\partial(T_{\textrm{\tiny DCFT}}\,I^{\textrm{\tiny
cold}}_{\textrm{\tiny E}})}{\partial T_{\textrm{\tiny DCFT}}}=0\,.\end{split}$
(296)

As expected, the entropy vanishes to leading order in a large $c$ expansion,
while the energy adopts a Casmir-like form. That is, $E^{\rm cold}$ is purely
a function of the boundary geometry, i.e., , $L_{1}$ and $L_{2}$, but
independent of the temperature. Further, the energy is negative since
$m_{i}<0$.

Two simple cases, which we might examine further, are: $m_{1}=m_{2}=m$ and
$|m_{1}|\ll|m_{2}|$. In the first case, we see from eq. (295) that
$L_{1}=L_{2}=\pi\ell/\sqrt{|m|}$, which agrees with eq. (72) in section 2.4
for the symmetric boundary. Further, as expected then, the action reduces to
the same expression found in eq. (73) and the Casimir form of the energy
becomes evident with

$E^{\rm cold}=-\frac{\pi\,c}{6}\,\frac{1}{L_{1}+L_{2}}\,.$ (297)

We note that this is the precise result that does not rely on the small $x$
limit.

In the regime $|m_{1}|\ll|m_{2}|$, we find from eq. (295) that to leading
order

$L_{1}\simeq\frac{2\pi\ell}{\sqrt{|m_{1}|}}\,,\qquad
L_{2}\simeq\frac{2\pi\ell}{\sqrt{|m_{2}|}}\left(1-\frac{1}{\sqrt{2}}\right)\,.$
(298)

Hence $L_{2}\ll L_{1}$ and we have arrived at the fusion limit considered in
section 2.3. Using the above expressions (298) to simplify the energy in eq.
(296), we arrive at

$E^{\rm
cold}\simeq-\frac{\pi\,c}{6}\left[\frac{1}{L_{2}}\left(1-\frac{1}{\sqrt{2}}\right)^{2}+\frac{1}{L_{1}}+\mathcal{O}\left(\frac{L_{2}}{L_{1}^{2}}\right)\right]\,.$
(299)

Comparing this expression with $E^{\rm cold}$ in eq. (66), we see that the two
energies agree when we substitute
$k_{2}\simeq\frac{\pi}{\sqrt{2}}+\mathcal{O}(x)$ from eq. (55).

Hence with these two simple limits, we confirm the present analysis for the
large tension limit by comparing to our previous results in sections 2.3 and
2.4. However, in principle, we have access to the thermodynamic behaviour for
general configurations with arbitrary values of $L_{1}$ and $L_{2}$. For
example, one could easily study small deviations from the symmetric
configuration, i.e., $m_{1}-m2=\delta m\ll 1$, in this regime. One could also
evaluate corrections in a small $x$ expansion.

Warm phase ($m_{1}>0$, $m_{2}<0$): The renormalized Euclidean action is now
represented by eq. (44) as:

$I^{\textrm{\tiny warm}}_{\textrm{\tiny E}}=\frac{c}{24\pi T_{\textrm{\tiny
DCFT}}}\left((L_{1}-2\ell\,\Delta\phi_{1}^{\textrm{\tiny
Hor}})\,m_{1}+L_{2}\,m_{2}\right)\,\frac{1}{\ell^{2}}\,.$ (300)

We need to determine $m_{i}$ and $\Delta\phi_{1}^{\text{Hor}}$ in terms of the
boundary data. In this scenario, a BTZ black hole exists in region
$\mathcal{S}_{1}$, and hence $m_{1}$ is determined by

$\frac{m_{1}}{\ell^{2}}=(2\pi\,T_{\textrm{\tiny DCFT}})^{2}\,,$ (301)

as in eq. (36). Meanwhile, $m_{2}$ is determined using eq. (294)

$L_{2}=2\ell\left(\frac{\pi}{\sqrt{-m_{2}}}-\Delta\phi_{2}\right)\,,$ (302)

as in the cold phase. Lastly, $\Delta\phi_{1}^{\text{Hor}}$ is determined by
eq. (38), resulting in

$\ell\,\Delta\phi_{1}^{\textrm{\tiny Hor}}=L_{1}+2\ell\,\Delta\phi_{1}\,.$
(303)

Using standard thermodynamic relations, one finds the energy and entropy are
given by

$\displaystyle E^{\rm warm}=-T_{\textrm{\tiny DCFT}}^{2}\frac{\partial
I^{\textrm{\tiny warm}}_{\textrm{\tiny E}}}{\partial T_{\textrm{\tiny DCFT}}}$
$\displaystyle=$ $\displaystyle\frac{c}{6}\left(\pi\,T_{\textrm{\tiny
DCFT}}^{2}\,(L_{1}+4\ell\,\Delta\phi_{1})+\frac{L_{2}}{4\pi\,}\,\frac{m_{2}}{\ell^{2}}\right)$
$\displaystyle\quad+\frac{c}{6}\left(4\pi\ell\,T_{\textrm{\tiny
DCFT}}^{3}\,\frac{\partial\Delta\phi_{1}}{\partial T_{\textrm{\tiny
DCFT}}}-\frac{L_{2}\,T_{\textrm{\tiny DCFT}}}{4\pi\,\ell^{2}}\,\frac{\partial
m_{2}}{\partial T_{\textrm{\tiny DCFT}}}\right)$ $\displaystyle S^{\rm
warm}=-\frac{\partial(T_{\textrm{\tiny DCFT}}\,I^{\textrm{\tiny
warm}}_{\textrm{\tiny E}})}{\partial T_{\textrm{\tiny DCFT}}}$
$\displaystyle=$ $\displaystyle\frac{\pi\,c}{3}\,T_{\textrm{\tiny
DCFT}}\,(L_{1}+4\ell\,\Delta\phi_{1})\,.$
$\displaystyle\quad+\frac{c}{6}\left(4\pi\ell\,T_{\textrm{\tiny
DCFT}}^{2}\,\frac{\partial\Delta\phi_{1}}{\partial T_{\textrm{\tiny
DCFT}}}-\frac{L_{2}}{4\pi\,\ell^{2}}\,\frac{\partial m_{2}}{\partial
T_{\textrm{\tiny DCFT}}}\right)$

Hence we see that region $\mathcal{S}_{1}$ contributes a thermal energy and
entropy (proportional to $L_{1}$) that is characteristic of the deconfined
phase of the CFT. The two contributions with $m_{2}$ and $\Delta\phi_{1}$ are
more complicated because generally these parameters now depend on the
temperature through the appearance of $m_{1}$ in eqs. (283) and (284).

One can make these observations more explicit but recalling some aspects of
the warm phase from section 2.2. From the discussion there and the phase
diagram in figure 6, we expect $L_{2}\ll L_{1}$ to be in this phase. That is,
we are in the the fusion limit described in section 2.3. Further to be below
the phase transition between the hot and warm phases, we have
$T_{\textrm{\tiny DCFT}}L_{2}<C_{\textrm{\tiny WH}}$ but in the large tension
limit, eq. (58) shows that $C_{\textrm{\tiny WH}}\sim c/\log g\ll 1$ and hence
we have $T_{\textrm{\tiny DCFT}}L_{2}\ll 1$. However, to be above the cold-
warm transition, we must have $T_{\textrm{\tiny DCFT}}L_{1}\gtrsim 1$ – see
discussion around eq. (59). That is, we must satisfy three constraints to be
in the warm phase

$L_{1}\gg L_{2}\,,\qquad T_{\textrm{\tiny DCFT}}L_{2}\ll 1\,,\qquad
T_{\textrm{\tiny DCFT}}L_{1}\gtrsim 1\,.$ (305)

Now expanding the expressions in eq. (C) to leading order for small
$T_{\textrm{\tiny DCFT}}L_{2}$, we find

$\displaystyle E^{\rm warm}$ $\displaystyle=$
$\displaystyle\frac{\pi\,c}{6}\left(T_{\textrm{\tiny
DCFT}}^{2}\,L_{1}-\frac{1}{L_{2}}\,\left(1-\frac{1}{\sqrt{2}}\right)^{2}+\frac{T_{\textrm{\tiny
DCFT}}^{2}\,L_{2}}{\sqrt{2}-1}+{\mathcal{O}}(T_{\textrm{\tiny
DCFT}}^{4}\,L_{2}^{3})\right)\,,$ (306) $\displaystyle S^{\rm warm}$
$\displaystyle=$ $\displaystyle\frac{\pi\,c}{3}\left(T_{\textrm{\tiny
DCFT}}\,L_{1}+\frac{T_{\textrm{\tiny
DCFT}}\,L_{2}}{\sqrt{2}-1}+{\mathcal{O}}(T_{\textrm{\tiny
DCFT}}^{3}\,L_{2}^{3})\right)\,,$ (307)

The first term in these expressions corresponds to the thermal energy and
entropy of the deconfined phase of the CFT in the $\mathcal{S}_{1}$ region of
length $L_{1}$ – see eq. (310) below. The second term in $E^{\rm warm}$
corresponds to a Casimir energy for the $\mathcal{S}_{2}$ region of length
$L_{2}$. We note that this term precisely matches the $1/L_{2}$ contribution
found in eq. (299) for the fusion limit in the cold phase. The terms
proportional to $L_{2}$ (and higher powers) are corrections indicating that
there is an interaction between the thermal phase in the $\mathcal{S}_{1}$
region and the Casimir phase in the $\mathcal{S}_{2}$ region. In fact, the
linear terms have the form of thermal contributions for a region of length
$L_{2}$, however, we note that the coefficient is larger than that expected
for the thermal CFT because of the factor $(\sqrt{2}-1)^{-1}\simeq 2.41$.
Examining the entropy, one finds that it is precisely the horizon entropy of
the black hole in the bulk $\mathcal{S}_{1}$ region, as expected – see eq.
(125) and also eq. (2.3) for the large tension values of $k_{1}$ and $k_{2}$.
A final observation is that in contrast to the hot phase below, there is no
defect contribution to the entropy here. Hot phase ($m_{i}>0$): In this case,
we have

$\frac{m_{1}}{\ell^{2}}=\frac{m_{2}}{\ell^{2}}=(2\pi\,T_{\textrm{\tiny
DCFT}})^{2}\,,$ (308)

from eq. (45). With this and other simplifications, the action reduces to

$I_{\textrm{\tiny E}}^{\rm hot}=-\frac{\pi\,c}{6}T_{\textrm{\tiny
DCFT}}\left(L_{1}+L_{2}\right)-4\log g$ (309)