USTC-ICTS/PCFT-23-36

Cutoff brane vs the Karch-Randall brane: the fluctuating case

Jiong Lin1,2 $\ast$, Yizhou Lu3 $\ast$ and Qiang Wen4 $\dagger$

1 Interdisciplinary Center for Theoretical Study, University of Science and
Technology of China, Hefei, Anhui 230026, China

2 Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

3 Department of Physics, Southern University of Science and Technology,
Shenzhen 518055, China

4  Shing-Tung Yau Center and School of Physics, Southeast University, Nanjing
210096, China

$\ast$ These authors contribute equally

$\dagger$ Corresponding to<EMAIL_ADDRESS>
## Abstract

Recently, certain holographic Weyl transformed CFT2 is proposed to capture the
main features of the AdS3/BCFT2 correspondence [1, 2]. In this paper, by
adapting the Weyl transformation, we simulate a generalized AdS/BCFT set-up
where the fluctuation of the Karch-Randall (KR) brane is considered. In the
gravity dual of the Weyl transformed CFT, the so-called cutoff brane induced
by the Weyl transformation plays the same role as the KR brane. Unlike the
non-fluctuating configuration, in the $2d$ effective theory the additional
twist operator is inserted at a different places, compared with the one
inserted on the brane. Though this is well-understood in the Weyl transformed
CFT set-up, it is confusing in the AdS/BCFT set-up where the effective theory
is supposed to locate on the brane. This confusion indicates that the KR brane
may be emergent from the boundary CFT2 via the Weyl transformations.

We also calculate the balanced partial entanglement (BPE) in the fluctuating
brane configurations and find it coincide with the entanglement wedge cross-
section (EWCS). This is a non-trivial test for the correspondence between the
BPE and the EWCS, and a non-trivial consistency check for the Weyl transformed
CFT set-up.

###### Contents

  1. 1 Introduction
  2. 2 A brief introduction to PEE and BPE in island phases
    1. 2.1 Partial entanglement entropy
    2. 2.2 Balanced partial entanglement
    3. 2.3 Ownerless island and generalized ALC formula
  3. 3 AdS/BCFT with fluctuating KR brane and its simulation as a Weyl transformed CFT
    1. 3.1 JT gravity from the fluctuation of the KR brane
    2. 3.2 Entanglement entropy with fluctuating KR brane
    3. 3.3 KR brane vs the cutoff brane from Weyl transformations
  4. 4 Entanglement wedge cross-sections with fluctuating brane
    1. 4.1 No-island phase
    2. 4.2 Island phases
  5. 5 BPE in Weyl transformed CFT
    1. 5.1 Adjacent $AB$ with island
    2. 5.2 Disjoint $AB$ with island
    3. 5.3 BPE from minimizing the crossing PEE
  6. 6 Discussion
  7. A Cutoff brane of Weyl transformed CFT
  8. B A trick to calculate EWCS for phase D2b

##  1 Introduction

In AdS/CFT correspondence [3], the entanglement entropy $S_{A}$ for a region
$A$ on the boundary CFT is proposed to be captured by the area of the minimal
surface $\mathcal{E}_{A}$ homologous to $A$ in the AdS bulk. This is the
famous Ryu-Takayanagi (RT) formula [4, 5]

$S_{A}=\frac{\text{Area}(\mathcal{E}_{A})}{4G}.$ (1)

Taking account of the quantum corrections from bulk fields, the RT formula was
then refined to the quantum extremal surface (QES) formula [6, 7]. The QES
formula can be applied to compute the entanglement entropy of the Hawking
radiation after Page time [8, 9], and remarkably a region in the black hole
interior could be regarded as part of the Hawking radiation. Such a region is
called the entanglement island, and QES formula in this context was refined to
be the island formula [10, 11, 12]111See [13, 2, 14, 15, 1, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] for other recent
processes in island.,

$\displaystyle
S(\mathcal{R})=\min\text{ext}_{\text{Is}(\mathcal{R})}\left[\frac{\text{Area}(\partial\text{Is}(\mathcal{R}))}{4G_{N}}+S_{\rm
bulk}(\mathcal{R}\cup\text{Is}(\mathcal{R}))\right]\,,$ (2)

Where $\mathcal{R}$ represent the Hawking radiatin and
$\mathrm{Is}(\mathcal{R})$ is the corresponding entanglement island. In [11,
12], the island formula was derived from the gravitational path integral,
where a new geometric saddle called the replica wormhole dominates the path
integral after the Page time.

The island formula can be realized in the 2$d$ effective description of
AdS3/BCFT2 set-ups [21, 34] via Karch-Randall braneworld holography [35, 36,
37]. By braneworld holography, one gets the induced gravity on the brane. When
the brane locates at the constant $\rho$ slice where $\rho$ is the hyperbolic
angle, the induced gravity is topological [21, 34]. If we let the brane
fluctuate in the bulk and keep the first order fluctuation
$\rho=\rho_{0}+\delta\rho$, then the localized gravity on the brane coupled to
the scalar degree describing the fluctuation is precisely the JT gravity [38,
39, 40]. The induced gravity on the brane connected with a non-gravitational
CFT bath with transparent boundary conditions, provides a 2$d$ effective
description of AdS/BCFT. Then by applying island formula on this $2d$
boundary, the entanglement entropy is exactly the same as the one from
$3d$-bulk RT-like formula.

Very recently, it was proposed in [1] that the island formula can be
understood from a pure quantum information perspective based on certain non-
trivial constraints on the Hilbert space. More explicitly the constraints that
induce entanglement islands should be understood as projecting out certain
states in the Hilbert space, such that for all the states remaining in the
reduced Hilbert space, there exists a mapping from the state of the subset
$\mathcal{R}$ to the state of the subset $I$, which we call a coding relation.
In other words, with the Hilbert space properly reduced, given the state of
$\mathcal{R}$ one can determine the state of $I$ through the coding relation.
It is important the stress that, the constraints are not just imposed on one
state of the system, they are imposed to the whole Hilbert space, such that
for any state reduced Hilbert space, it should be confined in this reduced
Hilbert spaces under any operations or evolution. Under such constraints,
spacelike separated degrees of freedom on a Cauchy slice could depend on each
other, which is a relation goes beyond causality.

We call a system _self-encoded_ if the Hilbert space is properly reduced
following the above described constraints. Consider one subregion
$\mathcal{R}$ in the system, and assume that the state of another subregion
$I$ is totally encoded in the state of $\mathcal{R}$, the coding relation can
be simply summarized by the following equation,

$\displaystyle\ket{i}_{\mathcal{R}}\Rightarrow\ket{f(i)}_{I}\,.$ (3)

In the path integral formalism, when we compute the elements of the reduced
density matrix of $\mathcal{R}$, we cut the region $\mathcal{R}$ open and set
different boundary conditions on the open edges. In this case, due to the
self-encoding property, we should simultaneously cut $I$ open and set boundary
conditions following the coding relation (3). One of the important consequence
is that, we can only trace the degrees of freedom in the complement of
$\mathcal{R}\cup I$. The other is that, additional twist operators are
inserted at the boundary of $I$. Eventually we arrive at the following formula
for the entanglement entropy of $\mathcal{R}$,

$\displaystyle S({\mathcal{R}})$ $\displaystyle=\tilde{S}({\mathcal{R}\cup
I})$ (4) $\displaystyle=\frac{\mathrm{Area}(\partial
I)}{4G_{N}}+\tilde{S}_{\text{bulk}}(\mathcal{R}\cup I)\,.$ (5)

In the above equation $\tilde{S}(A)$ represents the von Neumann entropy of the
reduced density matrix calculated by tracing out the degrees of freedom in the
complement of $A$. Like the configurations where the island formula (2)
applies, in the second equality we have assumed that the region $I$ is settled
in gravitational background. Hence the von Neumann entropy consists of two
parts, the gravitational contribution $\frac{\mathrm{Area}(\partial I)}{4G}$
and the bulk entanglement entropy $\tilde{S}_{\text{bulk}}(\mathcal{R}\cup I)$
[6, 41]. As was pointed out in [1], the formula (4) coincide with the island
formula (2), with the only difference that (2) is an optimization while (4) is
fixed by the coding relation (3). The difference is understandable since the
coding relation that result in the formula (2) should be much more complicated
than the simple relation (3). Later we also call $I$ the entanglement island
of $\mathcal{R}$ and denote it as Is$(\mathcal{R})$, and refer to the formulas
(2) and (4) the island formula $I$ and island formula $II$ respectively.

In [1] it was conjectured that, the island formula $I$ is a special
application of the island formula $II$ in gravitational systems. This
conjecture indicates that gravitational theories are self-encoded with the
Hilbert space vastly reduced under certain constraints. Although such
constraints are not well understood so far, there are many clues supporting
the statement that, the Hilbert space of an effective theory of gravity is
vastly reduced compared with a QFT in curved spacetime (see [1] for more
information). Inspired by the self-encoding property and the island formula
$II$, the authors of [1] proposed that, a holographic CFT2 under certain Weyl
transformation which introduces finite cutoff by erasing the UV physics of the
theory in certain regions, could be a perfect simulation for the Hilbert space
reduction of the gravity. More explicitly, the so-called cutoff brane in the
Weyl transformed CFT2 set-up plays the role of the KR (or end of world) brane
in the AdS3/BCFT2 set-up [42], and the entanglement structure of the Weyl
transformed CFT2 perfectly reproduces the island formula $I$ in the AdS/BCFT
set-up.

For a given region $A$, if the KR brane in the bulk fluctuates, the RT surface
$\mathcal{E}_{A}$, entanglement wedge $\mathcal{W}_{A}$ and the entanglement
wedge cross-section (EWCS)222Note that the cross-section of the entanglement
wedge may have multiple saddle points. In this case the EWCS represents the
saddle with the minimal area. of $\mathcal{W}_{A}$ should change accordingly.
Recently the holographic dual of the EWCS attracts considerable attention.
There is a zoo of quantum information quantities that are proposed to be the
quantum information dual of the EWCS, including the entanglement of
purification [43, 44, 45], the entanglement negativity [46, 47, 48], the
reflected entropy [49], the odd entropy [50], the differential purification
[51], the entanglement distillation [52, 53] and balanced partial entanglement
entropy (BPE) [54, 55, 56, 57]. The BPE has lots of advantages, for example it
is easy to compute, can be defined under different purifications, can be
defined in non-holographic systems and reduces to the reflected entropy in
canonical purification (see the introduction section of [55]). The
correspondence between the BPE and the EWCS has also passed highly non-trivial
tests in both static or covariant configurations in AdS3/CFT2 [54, 55], in
holographic CFT2 with gravitational anomaly [55], and even in 3d flat
holography [58, 59, 56, 57].

Quite recently,the PEE and BPE were studied in configurations with
entanglement islands [2]. By taking into account the island contributions, the
authors gave a generalized formula to construct the PEE and compute the BPE.
Here the so-called ownerless island region, which lies inside the island
Is$(AB)$ of $AB$ but outside Is$(A)\cup$Is$(B)$, plays a crucial role.
Remarkably, we find that under different assignments for the ownerless island,
we get different BPEs, which exactly correspond to the different saddles for
the cross-section of the entanglement wedge $\mathcal{W}_{AB}$. The
calculations are performed in several set-ups, including the Weyl transformed
CFT [1] (coupled to gravity) and a generalized AdS/BCFT model with a defected
theory localized on the brane [21].

In this paper we will test the correspondence between BPE and EWCS in a
modified version of the AdS/BCFT, where the KR (or EoW) brane in the bulk
fluctuates. It is interesting that the effective theory for the scalar freedom
that characterizes the first order fluctuation of the brane is exactly the JT
gravity [38, 39, 40] (see also [60, 61] for studies with fluctuating KR
brane). We will simulate this configuration using the Weyl transformed CFT2
and adapt the Weyl transformations to the fluctuations of the brane. An import
observation we will make is that, unlike the non-fluctuating configuration,
the twist operators are inserted at different places respectively in the Weyl-
transformed CFT at the asymptotic boundary and the KR brane in the bulk. And
only the inserting position on the Weyl-transformed CFT2 will generate the
right BPE that coincide with the EWCS. This strongly indicates that the KR
brane could be emergent from the Weyl transformed CFT2 at the asymptotic
boundary.

The paper is organized as follows. In Sec. 2, we start with a brief
introduction to the concept of PEE and BPE, and their construction in island
phases. In Sec. 3, we will first give a quick review on the JT gravity from
fluctuating Karch-Randall brane in the AdS/BCFT set-up. Then the main task of
this section is devoted to understanding why the island boundary point no
longer corresponds to the intersection point between the RT-like surface and
the brane in light of Weyl transformed CFT. Sec. 4 and Sec. 5 are devoted to
study the correspondence between BPE and EWCS in the AdS/BCFT set-up or Weyl
transformed CFT set-up with a fluctuating brane. We will start with the
derivation of the area of EWCS in Sec.4. We conclude our results in Sec.6.

##  2 A brief introduction to PEE and BPE in island phases

In this section, we give a brief introduction to PEE and BPE, and their
construction in island phases [2]. For the union of sets, we will suppress
"$\cup$" symbol for simplicity, i.e. we define $AB\equiv A\cup B$.

###  2.1 Partial entanglement entropy

The partial entanglement entropy (PEE) [62, 63, 64, 65, 66, 67, 68]333See [69,
70, 71, 72, 73, 74] for other recent progress on PEE. is a two-body measure of
entanglement featured by the property of additivity. It is usually written as
$\mathcal{I}(A,B)$ for two non-overlapping spacelike separated subsystems $A$
and $B$. This is the two-body correlation representation for the PEE. Note
that, we should not mix between the PEE $\mathcal{I}(A,B)$ and the mutual
information $I(A,B)$. Although the explicit definition for the PEE based on
reduced density matrix is still not clear, in many configurations the PEE can
be determined by its physical requirements, which including all the
requirements satisfied by the mutual information $I(A,B)$ and the additional
requirement of additivity [62, 66], i.e.

  1. 1.

Additivity: $\mathcal{I}(A,B\cup C)=\mathcal{I}(A,B)+\mathcal{I}(A,C)$;

  2. 2.

Permutation symmetry: $\mathcal{I}(A,B)=\mathcal{I}(B,A)$;

  3. 3.

Normalization: $\mathcal{I}(A,\bar{A})=S_{A}$;

  4. 4.

Positivity: $\mathcal{I}(A,B)>0$;

  5. 5.

Upper bounded: $\mathcal{I}(A,B)\leq\text{min}\\{S_{A},S_{B}\\}$;

  6. 6.

$\mathcal{I}(A,B)$ should be Invariant under local unitary transformations
inside $A$ or $B$;

  7. 7.

Symmetry: For any symmetry transformation $\mathcal{T}$ under which
$\mathcal{T}A=A^{\prime}$ and $\mathcal{T}B=B^{\prime}$, we have
$\mathcal{I}(A,B)=\mathcal{I}(A^{\prime},B^{\prime})$.

It has been demonstrated that [75, 66], the above requirements has unique
solution for Poincaré invariant theories, and furthermore, for conformal field
theory the formula of the solution can be determined by the requirements.
There are also other prescriptions to construct the PEE satisfying the above
requirements, like the Gaussian formula applies for Gaussian states [62],
geometric construction applied to holographic configurations with a local
modular Hamiltonian [63, 67], and the so-called additive linear combination
(ALC) proposal [63, 67] applied to generic systems settled on a chain (or
line) with a definite order. The ALC proposal will be the workhorse in this
paper, which is given in the following.

  * •

The ALC proposal [63, 67, 66]:

Consider a boundary region $A$ and partition it into three non-overlapping
subregions $A=\alpha_{L}\cup\alpha\cup\alpha_{R}$, where $\alpha$ is some
subregion inside $A$ and $\alpha_{L}$ ($\alpha_{R}$) denotes the regions left
(right) to it. On this configuration, the claim of the ALC proposal is the
following:

$\displaystyle
s_{A}(\alpha)=\mathcal{I}(\alpha,\bar{A})=\frac{1}{2}\left(S_{\alpha_{L}\cup\alpha}+S_{\alpha\cup\alpha_{R}}-S_{\alpha_{L}}-S_{\alpha_{R}}\right)\,.$
(6)

By additivity and permutation symmetry, the PEE $\mathcal{I}(A,B)$ can be
decomposed into two-point PEEs $\mathcal{I}(\textbf{x},\textbf{y})$ , i.e.

$\mathcal{I}\left(A,B\right)=\int_{A}\mathrm{d}\sigma_{\textbf{x}}\int_{B}\mathrm{d}\sigma_{\textbf{y}}~{}\mathcal{I}(\textbf{x},\textbf{y})\,,\quad\textbf{x}\in
A\,,\quad\textbf{y}\in B\,,$ (7)

which means all the PEE can be generated from the two point PEEs. In a
discrete chain, the number of the two-point PEEs equals to the number of all
connected intervals, which means that the information contained in all the
two-point PEEs is indeed a reformulation of the information of all the
entanglement entropy for connected intervals.

The concept of PEE originates from the study of the entanglement contour
$s_{A}(\textbf{x})$ [62], which is assumed to be a spatial density function of
entanglement entropy that captures the contribution from each local degree of
freedom to $S_{A}$. The quantity $s_{A}(A_{i})$ is then defined as the
collections of the entanglement contributions from a subregion $A_{i}$ of $A$
to the entanglement entropy $S_{A}$, i.e.

$s_{A}(A_{i})=\int_{A_{i}}s_{A}(\mathbf{x})\mathrm{d}\sigma_{\mathbf{x}},$ (8)

where $\sigma_{\mathbf{x}}$ is an infinitesimal subset. This is just the
contribution representation of the PEE,

$\mathcal{I}(A_{i},\bar{A})\equiv s_{A}(A_{i}),$ (9)

where $\bar{A}$ denotes the complement of $A$ that purifies $A$.

###  2.2 Balanced partial entanglement

The balanced partial entanglement entropy (BPE) a PEE that satisfies the
balanced requirements. For a system $A\cup B$ in a mixed state $\rho_{AB}$,
one can introduce an auxiliary system $A^{\prime}B^{\prime}$ to purify $AB$
such that the whole system $ABA^{\prime}B^{\prime}$ is in a pure state
$|\varphi\rangle$ and,

$\displaystyle\text{Tr}_{A^{\prime}B^{\prime}}\ket{\varphi}\bra{\varphi}=\rho_{AB}.$
(10)

The pure state on $ABA^{\prime}B^{\prime}$ is called a purification of
$\rho_{AB}$, which could be highly non-unique. Let us consider the simple
examples described in Fig.1, for each purification the balanced requirements
are summarized in the following [54]:

Figure 1:  BPE for adjacent and disjoint cases are shown on the left and right
respectively. $q,q_{1},q_{2}$ are balanced points. This figure is extracted
from [2].

  * •

Balance requirement: When $A$ and $B$ are adjacent, the contribution from $A$
to $S_{AA_{1}}$ should be equal to the contribution from $B$ to $S_{BB_{1}}$,
that is

$s_{AA_{1}}(A)=s_{BB_{1}}(B),\
\text{or}\quad\mathcal{I}(A,BB_{1})=\mathcal{I}(B,AA_{1}).$ (11)

This requirement is enough to determine the partition point in the purifying
system $A_{1}B_{1}$, which we call the balance point.

  * •

When $A$ and $B$ are non-adjacent, the balance requirements become

$s_{AA_{1}A_{2}}(A_{1})=s_{BB_{1}B_{2}}(B_{1}),\
s_{AA_{1}A_{2}}(A)=s_{BB_{1}B_{2}}(B),$ (12)

or

$\mathcal{I}(A,B_{1}B_{2})=\mathcal{I}(B,A_{1}A_{2}),\
\mathcal{I}(A_{1},BB_{2})=\mathcal{I}(B_{1},AA_{2}),$ (13)

which are enough to determine the two partition points of the purifying system
$A_{1}B_{1}A_{2}B_{2}$. Since $S_{AA_{1}A_{2}}=S_{BB_{1}B_{2}}$,
$s_{AA_{1}A_{2}}(A_{2})=s_{BB_{1}B_{2}}(B_{2})$ is automatically satisfied
provided the satisfaction of the above requirements.

  * •

Note that, it is possible that the solution to the balance requirements could
be non-unique. In such cases, we should choose the solution that minimizes the
BPE, which will be defined soon.

Provided the balanced requirements are fulfilled, the BPE is then defined as

$\text{BPE}(A,B)=s_{AB}(B)|_{\rm balanced}=s_{AB}(A)|_{\rm balanced}\,.$ (14)

Since we have $\mathcal{I}(A,B^{\prime})=\mathcal{I}(A^{\prime},B)$ at the
balance point, the BPE can also be expressed as

$\text{BPE}(A,B)=\mathcal{I}(A,B)+\frac{(\mathcal{I}(A,B^{\prime})+\mathcal{I}(A^{\prime},B))|_{\rm
balanced}}{2}.$ (15)

It is important to note that [56], the summation
$\mathcal{I}(A,B^{\prime})+\mathcal{I}(A^{\prime},B)$ which we call the
crossing PEE is minimized under the balanced point. This indicates that the
BPE can be defined via an optimization problem. It is also interesting to note
that, when $A$ and $B$ are adjacent, this minimal crossing PEE gives half of
the lower bound of Markov gap [76, 54, 77], which is a universal constant
$(c/6)\log 2$.

###  2.3 Ownerless island and generalized ALC formula

Now we generalize our construction of the PEE and BPE to configurations with
entanglement islands [2]. Here we just list the basic elements we need to
carry out the computations. One should consult [2] for more details about the
physical reasoning behind our generalization. Let us start with the region
$AB$ and the island regions Is$(AB)$, Is$(A)$ and Is$(B)$. According to [2],
the degrees of freedom in $\mathrm{Is}(AB)$ is not independent, and contribute
to the entanglement entropy $S_{AB}$ calculated by the island formula. The
PEE, for example $s_{AB}(A)$, should contain the contribution from the island
region, and it is clear that we should assign the contribution from
$\mathrm{Is}(A)$ to $s_{AB}(A)$ and similarly assign $\mathrm{Is}(B)$ to
$s_{AB}(B)$. Nevertheless, there are scenarios with regions included in
$\mathrm{Is}(AB)$ but outside $\mathrm{Is}(A)\cup\mathrm{Is}(B)$, which should
also be assigned to $s_{AB}(A)$ or $s_{AB}(B)$. In other words, when
$\mathrm{Is}(AB)\neq\emptyset$ and
$\mathrm{Is}(AB)\supset\mathrm{Is}(A)\cup\mathrm{Is}(B)$, we define the
ownerless island region to be

$\displaystyle\text{Io}(AB)=\mathrm{Is}(AB)/(\mathrm{Is}(A)\cup\mathrm{Is}(B))\,.$
(16)

The ownerless island should be further divided into two parts

$\displaystyle\text{Io}(AB)=\text{Io}(A)\cup\text{Io}(B)\,,$ (17)

which are assigned to $s_{AB}(A)$ and $s_{AB}(B)$ respectively. We will
specify the division of Io$(AB)$ later. All in all, we define the _generalized
islands_

$\displaystyle\mathrm{Ir}(A)=\mathrm{Is}(A)\cup\text{Io}(A),\qquad\mathrm{Ir}(B)=\mathrm{Is}(B)\cup\text{Io}(B)\,,$
(18)

and assign them to $A$ and $B$ respectively when calculating the PEEs. We can
classify the assignment into three classes.

  * •

Class 1: When Is$(AB)$=Is$(A)$=Is$(B)$=$\emptyset$, we have

$\displaystyle\mathrm{Ir}(A)=\mathrm{Ir}(B)=\emptyset\,.$ (19)

  * •

Class 2: When Is$(AB)$=$\mathrm{Is}(A)\cup\mathrm{Is}(B)\neq\emptyset$, we
have

$\displaystyle\text{Io}(A)=\text{Io}(B)=\text{Io}(AB)=\emptyset,\qquad\mathrm{Ir}(A)=\mathrm{Is}(A),\qquad\mathrm{Ir}(B)=\mathrm{Is}(B)\,.$
(20)

  * •

Class 3: When we have non-trivial ownerless island region, then we have (18).

Now we turn to the computation of the PEEs and denote
$C\equiv\overline{AB\cup\text{Is}(AB)}$ for convenience. After taking into
account the contributions from the islands, we should have [2],

$s_{AB}(A)=\mathcal{I}(A\cup\text{Ir}(A),C),\quad
s_{AB}(B)=\mathcal{I}(B\cup\text{Ir}(B),C).$ (21)

A key step to compute the PEEs (21) is to write the right hand side of (21) as
a linear combination of the PEEs that can be written in this form
$\mathcal{I}(\gamma,\bar{\gamma})$, i.e. a PEE between a single interval and
its complement. The PEEs $\mathcal{I}(\gamma,\bar{\gamma})$ are then
calculated following the basic proposal in [2],

$\textit{Basic
proposal}:~{}~{}\mathcal{I}(\gamma,\bar{\gamma})=\tilde{S}_{\gamma}=\tilde{S}_{[-a,b]},\qquad\gamma:[-a,b]\,,$
(22)

which are just the two-point function of twist operators inserted at the
endpoints of $\gamma$. Such two-point functions are given by applying a
special type of Weyl transformations to the two-point functions in normal
CFT2. Based on the basic proposal (22), we can calculate the PEE
$\mathcal{I}(\gamma,\bar{\gamma})$ for disconnected
$\gamma=A\cup\text{Ir}(A)$, where $\text{Ir}(A)=[-d,-c]$, $A=[a,b]$ and
$a,b,c,d>0$ (see [2] for the demonstration),

$\displaystyle\mathcal{I}(A\,\text{Ir}(A),\overline{A\,\text{Ir}(A)})=\tilde{S}_{[-d,-c]\cup[a,b]}=\tilde{S}_{[-c,a]}+\tilde{S}_{[-d,b]}\,.$
(23)

Using this proposal and the additivity property, the PEE (21) is just given by

$\begin{split}s_{AB}(A)=&\mathcal{I}(A\cup\text{Ir}(A),C)\\\
=&\frac{1}{2}\left[\mathcal{I}(A\text{Ir}(A)B\text{Ir}(B),C)+\mathcal{I}(A\text{Ir}(A),B\text{Ir}(B)C)-\mathcal{I}(B\text{Ir}(B),A\text{Ir}(A)C)\right]\\\
=&\frac{1}{2}\left[\tilde{S}_{A\text{Ir}(A)B\text{Ir}(B)}+\tilde{S}_{A\text{Ir}(A)}-\tilde{S}_{B\text{Ir}(B)}\right].\end{split}$
(24)

This is our _generalized ALC formula_ [2] in island phases, which is just the
ALC formula (6) with the replacement
$S_{\gamma}\Rightarrow\tilde{S}_{\gamma\text{Ir}(\gamma)}$ applied to each
term. Similarly, when $A$ is sandwiched by $A_{1}$ and $A_{2}$, we have

$s_{A_{1}AA_{2}}(A)=\frac{1}{2}\left[\tilde{S}_{A_{1}\text{Ir}(A_{1})A\text{Ir}(A)}-\tilde{S}_{A_{1}\text{Ir}(A_{1})}+\tilde{S}_{A\text{Ir}(A)A_{2}\text{Ir}(A_{2})}-\tilde{S}_{A_{2}\text{Ir}(A_{2})}\right].$
(25)

The remaining problem is the division of the ownerless island region
$\text{Io}(AB)=\text{Io}(A)\cup\text{Io}(B)$. In [2], the authors considered
the AdS/BCFT set-up and its simulation via a holographic Weyl transformed
CFT2. They found that different assignment for the ownerless island region
lead to different BPEs, which exactly correspond to different saddles of the
EWCS. Then according to the minimum requirement, we should choose the
assignment that gives us the minimal BPE. It seems that, the division of the
ownerless island depend on the phase structure of the EWCS. In this work, we
will study this correspondence in a more generic AdS/BCFT set-up, where we
allow the brane to fluctuate in the bulk. Similarly we simulate this set-up
via a Weyl transformed CFT2 with the Weyl transformation modified to adapt the
fluctuation of the brane.

##  3 AdS/BCFT with fluctuating KR brane and its simulation as a Weyl
transformed CFT

###  3.1 JT gravity from the fluctuation of the KR brane

Let us start with a quick look at the idea of JT gravity from dimensional
reduction [38, 39, 40]. Consider the AdS/BCFT correspondence with the bulk
action,

$\displaystyle I_{\text{AdS}}$ $\displaystyle=\frac{1}{16\pi
G_{N}}\int_{N}\sqrt{-g}(R-2\Lambda)+\frac{1}{8\pi
G_{N}}\int_{\mathcal{Q}}\sqrt{-h}(K-T),$ (26)

where $N$ denotes the bulk AdS spacetime and $\mathcal{Q}$ denotes the Karch-
Randall (KR) brane with the Neumann boundary condition imposed and $T$ is the
brane tension. It is convenient to write the bulk metric in terms of the
following coordinates,

$\displaystyle\mathrm{d}s^{2}=$
$\displaystyle\frac{\ell^{2}}{z^{2}}(-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}z^{2})$
(27) $\displaystyle=$
$\displaystyle\mathrm{d}\rho^{2}+\ell^{2}\cosh^{2}\frac{\rho}{\ell}\frac{-\mathrm{d}t^{2}+\mathrm{d}y^{2}}{y^{2}}\,,$
(28)

where $\ell$ is the AdS radius, which we will take to be unit $\ell=1$ in the
following, and the two sets of coordinates are related by,

$z=y\cosh^{-1}{\rho},\quad x=-y\tanh{\rho}.$ (29)

Here the brane is located at $\rho=\rho_{0}$, where $\rho_{0}$ is a constant
determined by the tension of the brane $\rho_{0}=\text{arctanh}~{}T$.

Let us divide the bulk into two wedges $W_{1}\cup W_{2}$ along the $\rho=0$
(see Fig. 2). By braneworld holography or dimensional reduction [35, 36, 37]
on wedge $W_{1}$ (see [21, 34] for example), we will get the induced gravity
on the KR brane, which is a topological gravity on the brane (see the upper
figure in Fig. 2). Later we just assume that the gravitational theory on the
brane is just the induced gravity. And the region $W_{2}$ is regarded as the
gravity dual of the CFT2 bath.

Figure 2:  Gravity on the KR brane $\mathcal{Q}$ via dimensional reduction on
$W_{1}$. The induced gravity on a brane at a fixed $\rho_{0}$ is topological,
while the induced gravity is JT gravity if we allow the brane to fluctuate.

Following [38, 39, 40], we let the KR brane fluctuate around constant
$\rho=\rho_{0}$ slice, and only keep the first order fluctuation. The brane
fluctuation $\delta\rho/\rho_{0}$ can be described by a scalar degree of
freedom, and results in a 2$d$ effective action (to the order
$\mathcal{O}(\delta\rho/\rho_{0})$) on the brane. This theory is exactly the
JT gravity coupled to a dilaton characterized by the scalar degree of freedom,

$\displaystyle I_{2d}$ $\displaystyle=\frac{\rho_{0}}{16\pi
G_{N}}\int\sqrt{-g^{(2)}}\frac{\delta\rho}{\rho_{0}}\left(R^{(2)}+\frac{2}{\cosh^{2}{\rho_{0}}}\right)+\frac{\rho_{0}}{16\pi
G_{N}}\int\sqrt{-g^{(2)}}R^{(2)}+\mathcal{O}\left(\frac{\delta\rho^{2}}{\rho_{0}^{2}}\right).$
(30)

The solution of dilaton on the brane ($y<0$) is given by,

$\frac{\delta\rho}{\rho_{0}}=-\frac{\bar{\phi}_{r}}{y}\,,\qquad y<0,$ (31)

where $\bar{\phi}_{r}$ is the renormalized boundary dilaton of JT gravity that
is taken as a constant [78]. Then the brane locates at (see the lower figure
in Fig. 2),

$\rho=\rho_{0}+\delta\rho=\rho_{0}\left(1-\frac{\bar{\phi}_{r}}{y}\right).$
(32)

Since the induced gravity on the fluctuating brane is JT gravity, later we
will also refer to this brane as the JT brane.

###  3.2 Entanglement entropy with fluctuating KR brane

Now we calculate the entanglement entropy in the $2d$ effective theory for an
interval $A=[0,L]$ via the island formula $I$. Assuming the island region to
be $\mathrm{Is}(A)=[-a,0)$, the island formula gives us

$S_{A}=\text{min~{}ext}_{a}\left[\frac{\text{Area}(\partial\mathrm{Is}(A))}{4G_{2}}+S_{\text{bulk}}([-a,L])\right]$
(33)

In the AdS/BCFT set-up, there is a holographic picture for the $2d$ effective
theory, where the bulk entanglement entropy $S_{\text{bulk}}([-a,L])$ can be
calculate by the area of the RT surface $\Gamma$ that anchors on the JT brane
(see Fig.3). Hence the island formula becomes444Here the
$S_{\text{bulk}}([-a,L])$ is calculated in a semi-classical way thus the bulk
entanglement entropy in AdS3 is ignored.

$\displaystyle
S_{A}=\text{min~{}ext}\left[\frac{\text{Area}(\partial\mathrm{Is}(A))}{4G_{2}}+\frac{\operatorname{Area}(\Gamma)}{4G_{3}}\right]$
(34)

In this paper, since we take the full JT gravity on the brane to be induced
from the KR reduction of the region $W_{1}$, the area term can be ignored [34,
79, 39]. On the effective theory side, $S_{\text{bulk}}([-a,L])$ is calculated
by a Weyl transformed two-point function of twist operators. So far, there are
two choices of the Weyl transformation, which give different
$S_{\text{bulk}}([-a,L])$.

#### Weyl factor 1

The first one is the Weyl transformation that relates the induced metric on
the brane with the flat metric. Up to the order $\mathcal{O}(\delta\rho)$, the
induced metric of the JT brane is given by,

$\begin{split}\mathrm{d}s^{2}_{\text{JT-
brane}}&={\cosh^{2}\rho}\left(\frac{-\mathrm{d}t^{2}+\mathrm{d}y^{2}}{y^{2}}\right)\\\
&\approx\left({\cosh^{2}\rho_{0}{+\delta\rho\sinh(2\rho_{0})}}\right)\left(\frac{-\mathrm{d}t^{2}+\mathrm{d}y^{2}}{y^{2}}\right)+\mathcal{O}(\delta\rho^{2})\\\
&=\Omega(y)^{-2}\frac{-\mathrm{d}t^{2}+\mathrm{d}y^{2}}{\epsilon^{2}}+\mathcal{O}(\delta\rho^{2}),\end{split}$
(35)

which is conformal flat in $(t,y)$ coordinates with the conformal factor

$\Omega^{-2}(y)=\frac{\epsilon^{2}}{y^{2}}\left({\cosh^{2}\rho_{0}+\delta\rho\sinh(2\rho_{0})}\right).$
(36)

Before we go ahead, we introduce a parameter

$\displaystyle\mu\equiv\frac{\rho_{0}\bar{\phi}_{r}}{6L}=-\frac{\delta\rho}{6}\frac{y}{L}.$
(37)

which represents the first order fluctuation of the brane as long as
$(y/L)\sim\mathcal{O}(1)$. In the rest of this paper we will only keep the
first order of the fluctuation, and equations with “$\approx$” only hold up to
the order $\mathcal{O}(\mu)$. Note that, compared with previous literature
[12, 40], we have introduced the additional parameter $\epsilon$ which
represents the UV cutoff of the boundary. Then the bulk entropy is given by

$\begin{split}S_{\text{bulk}}([-a,L])&=\frac{c}{6}\log\frac{(L+a)^{2}}{\Omega(-a)\epsilon^{2}}\\\
&\approx\frac{c}{6}\log\frac{(L+a)^{2}}{2a\epsilon}+\frac{c}{6}\log\left(2\cosh\rho_{0}\right)+\frac{cL\mu}{a}\tanh\rho_{0}.\end{split}$
(38)

Note that the subtle cutoff scale $\epsilon_{y}$ on the brane (see [21]) is
terminated by the Weyl factor $\Omega(-a)$ as (36) depend on $\epsilon$ and
diverges as $\epsilon\to 0$. The extremal point is given by

$a\approx a_{\rm bdy}=L+12L\mu\tanh\rho_{0},$ (39)

hence the entanglement entropy from the island formula $I$ is

$S_{A}\approx\frac{c}{6}\log\frac{2L}{\epsilon}+\frac{c}{6}\log\left(2\cosh\rho_{0}\right)+c\mu\tanh\rho_{0}.$
(40)

#### Weyl factor 2

The another choice for the Weyl factor (see [34, 1, 2] for examples) is
motivated from the role of the KR brane as a cutoff boundary of the AdS bulk.
The corresponding Weyl factor is given by (see next subsection)

$\displaystyle\Omega^{-2}(y)$
$\displaystyle=\exp\left[{-2\left(\log\left(\frac{2|y|}{\epsilon}\right)-\rho_{0}+\frac{\rho_{0}\bar{\phi}_{r}}{y}\right)}\right]$
(41)
$\displaystyle=\exp\left[-2\left(\log\frac{-2y}{\epsilon}-\rho_{0}+\frac{6\mu
L}{y}\right)\right]\quad y<0\,,$ (42)

which results in

$\begin{split}S_{\text{bulk}}([-a,L])&=\frac{c}{6}\log\frac{(L+a)^{2}}{\Omega(-a)\epsilon^{2}}\\\
&\approx\frac{c}{6}\log\frac{(a+L)^{2}}{2a\epsilon}+\frac{c}{6}\rho_{0}+\frac{cL\mu}{a}\,.\end{split}$
(43)

In this case, the extremal point is given by

$\displaystyle a\approx a_{\rm bdy}=L+12L\mu\,,$ (44)

hence the entanglement entropy from the island formula $I$ becomes

$\displaystyle
S_{A}\approx\frac{c}{6}\log\frac{2L}{\epsilon}+\frac{c}{6}\rho_{0}+c\mu\,.$
(45)

Figure 3: Entanglement entropy for $[0,L]$ from $3d$-bulk RT prescription.

#### Doubly holography calculation

From the $3d$-bulk viewpoint, the bulk entanglement entropy term in the island
formula $I$ is just given by the area of the RT surface $\Gamma$ (see Fig. 3)
that anchored on the brane vertically,

$S_{A}=\frac{\operatorname{Area}(\Gamma)}{4G_{N}}.$ (46)

The RT surface can be determined by minimizing the length of the geodesic (see
the blue curve in Fig. 3) that anchors on the brane at $y=-\tilde{a}$, i.e.
[40]

$\displaystyle\frac{\operatorname{Area}(\Gamma)}{4G_{N}}$
$\displaystyle=\text{Min}\left[\frac{c}{6}\log\left(\frac{\left(L+\tilde{a}\tanh\rho_{0}\right)^{2}+\tilde{a}^{2}\cosh^{-2}\rho_{0}}{\tilde{a}\epsilon\cosh^{-1}\rho_{0}}\right)+\frac{cL\mu}{\tilde{a}}\right].$
(47)

The extremal point is given by

$\tilde{a}\approx a_{\rm brane}=L+6L\left(1+\tanh\rho_{0}\right)\mu,$ (48)

where the subscript “brane” means that the extremal point is the point between
where the RT-like surface $\Gamma$ anchors on the KR brane. Finally, the
entanglement entropy is

$S_{A}\approx\frac{c}{6}\log\frac{2L}{\epsilon}+\frac{c}{6}\rho_{0}+c\mu.$
(49)

#### Two important observations

  * •

Firstly, it is obvious that, the holographic calculation (49) only matches
entanglement entropy (45) under the second Weyl transformation (41). While the
entanglement entropy (40) under the first Weyl transformation does not match
(49) even under the non-fluctuating limit $\mu\to 0$. It only match (49) in
the limit that the KR brane approaches the asymptotic boundary, such that the
cutoff $\epsilon_{y}$ on the brane is satisfy $0<\epsilon\ll\epsilon_{y}\ll 1$
(see [12]).

  * •

Secondly, the minimal point $a_{\rm brane}$ (48) for the RT curve differs from
both of the minimal points (39) and (44) of the island formula, i.e.

$\displaystyle a_{\rm brane}\neq a_{\rm bdy}\,.$ (50)

This is quite different from the case with non-fluctuating brane, where
$a_{\rm brane}=a_{\rm bdy}$ for both of the Weyl factors under the limit
$\mu\to 0$.

These issues will be well explained based on the simulation for the AdS/BCFT
via a Weyl transformed CFT2.

###  3.3 KR brane vs the cutoff brane from Weyl transformations

Figure 4: Cutoff sphere for Weyl transformed CFT. Note that the RT curves are
always normal to the cutoff sphere.

The Weyl transformed CFT simulation for the AdS/BCFT configuration was first
introduced in [1]. Such Weyl transformations depend on the UV cutoff and
erases the UV physics in certain region. It is assumed to be the operation
that vastly reduces the Hilbert space of the theory and hence induces the
self-coding property and the island formula for entanglement entropy [1].

Let us start with a holographic CFT2 with UV cutoff $\epsilon$. According to
the AdS/CFT, the metric background of the theory should be
$\mathrm{d}s^{2}=(-\mathrm{d}t^{2}+\mathrm{d}x^{2})/\epsilon^{2}$. Then we
perform a Weyl transformation on the theory, which changes the metric in the
following way

$\mathrm{d}s^{2}=\frac{-\mathrm{d}t^{2}+\mathrm{d}x^{2}}{\epsilon^{2}},\
\Rightarrow\
\mathrm{d}s^{2}=e^{2\varphi(x)}\left(\frac{-\mathrm{d}t^{2}+\mathrm{d}x^{2}}{\epsilon^{2}}\right).$
(51)

This effectively changes the cutoff in a position-dependent way,

$\epsilon\ \to\ e^{-\varphi(x)}\epsilon.$ (52)

Accordingly, the entanglement entropy for a single interval, for example
$A=[a,b]$, changes in a simple way

$\displaystyle
S_{A}=\frac{c}{3}\log\frac{(b-a)}{\epsilon}+\frac{c}{6}\varphi(a)+\frac{c}{6}\varphi(b)\,,$
(53)

which is just the original entanglement entropy adding the value of the scalar
field, which is negative, at the endpoints. Holographically, as shown in Fig.
4, the constant subjection of the entanglement entropy was understood as
inserting cutoff spheres with radius $|\varphi(x)|$ centered at the endpoints.
When computing the holographic entanglement entropy, we should exclude the
portion of the RT surface inside the cutoff spheres.

The particular Weyl transformation that captures the main features of the
AdS/BCFT with non-fluctuating brane is given by [1],

$\varphi(x)=\begin{cases}0,&\text{ if }\quad x>0\\\
-\log\left(\frac{2|x|}{\epsilon}\right)+\kappa_{1},&\ \text{if }\quad
x<0\end{cases}.$ (54)

In this case, the common tangent line (or the envelope surface) of all the
cutoff spheres is exactly a straight line locates at

$\displaystyle\rho=\kappa_{1}\,.$ (55)

Assuming that entanglement islands is allowed to emerge on the $x<0$
region,555This can be realized based on two set-ups. Firstly, one can couple
the left hand side of the CFT2 with gravity and assuming Neumann boundary
condition on this region (see the [34] for example), then the replica wormhole
configuration [12, 11] applies to this set-up. Secondly, entanglement island
emerges if we assume that the Weyl transformation induces the self-encoding
property of the system, see [1] for more details. the entanglement entropy for
regions on the right hand side $x>0$ could enter island phase according to the
island formula $I$. For regions in island phase, the RT surfaces are cut off
exactly at the cutoff brane, see Fig. 5. In other words the RT surface is
allowed to anchor on the cutoff brane, which plays the same role as the KR
brane in the AdS/BCFT correspondence. Since $\Omega(x)^{-2}=e^{2\varphi(x)}$,
this explains the choice of the Weyl factor 2 (41) in the AdS/BCFT set-up from
the cutoff point of view666Note that, here we have not included the
fluctuation yet, so the matching between $e^{2\varphi(x)}$ and (41) is not
exact. .

Figure 5: The cutoff brane in Weyl transformed CFT vs the KR brane at
$\rho=\rho_{0}$ in AdS/BCFT.

For example, if we apply island formula $I$ to calculate $S_{A}$ for the
interval $A=[0,L]$ in this Weyl transformed CFT and assuming the island region
$\text{Is}(A)=[-a,0)$, then we have

$\displaystyle
S_{A}=\text{min~{}ext}\left[\frac{c}{3}\log\frac{(L+a)}{\epsilon}+\frac{c}{6}\varphi(-a)+\frac{c}{6}\varphi(L)\right]\,.$
(56)

Plugging into (54), the above formula is minimized at

$\displaystyle a_{\rm bdy}=L,$ (57)

hence

$\displaystyle
S_{A}=\frac{c}{6}\log\frac{2L}{\epsilon}+\frac{c}{6}\kappa_{1}\,.$ (58)

This exactly matches the result in AdS/BCFT with $\rho_{0}=\kappa_{1}$. From
the bulk viewpoint, this entanglement entropy is exactly the area of the RT
surface connecting the two points at $x=L,-L$ and being cut off at the cutoff
brane. As shown in Fig. 5, since the center of this RT surface lies at the
origin, the intersection point between this RT surface and cutoff brane is at
$y=-a_{\rm brane}=-L$, thus $a_{\rm bdy}=a_{\rm brane}$. The coincidence
between the two set-ups can be straightforwardly extended to intervals
$A=[a,b]$ not including the origin.

Figure 6: The cutoff brane in Weyl transformed CFT vs the fluctuating KR brane
in AdS/BCFT. The dashed part of the blue curves is cut off when computing the
area of RT surface.

Now we simulate the AdS/BCFT configuration with fluctuating brane via the Weyl
transformed CFT2 with the Weyl transformations modified accordingly. Using
(29) and (32), the location of JT brane is given by the following parametric
equation,

$\begin{split}-x_{J}&=-|y|\tanh\left[\rho_{0}\left(1+\frac{\bar{\phi}_{r}}{|y|}\right)\right]\approx-|y|\tanh\rho_{0}-{\rho_{0}\bar{\phi}_{r}}\cosh^{-2}\left(\rho_{0}\right)+\mathcal{O}(\bar{\phi}_{r}^{2}),\\\
z_{J}&=|y|\cosh^{-1}\left[\rho_{0}\left(1+\frac{\bar{\phi}_{r}}{|y|}\right)\right]\approx|y|\cosh^{-1}\rho_{0}-{\rho_{0}\bar{\phi}_{r}}\cosh^{-1}\left(\rho_{0}\right)\tanh\left(\rho_{0}\right)+\mathcal{O}(\bar{\phi}_{r}^{2}).\end{split}$
(59)

Now we directly give the Weyl factor whose common tangent curve (or the
envelope surface) of all the cutoff spheres (or cutoff brane) matches the JT
brane,

$\varphi(x)=\begin{cases}0,&\text{ if }\quad x>0\\\
-\log\left(\frac{2|x|}{\epsilon}\right)+\kappa_{1}-\frac{\kappa_{2}}{x},&\
\text{if }\quad x<0\end{cases}.$ (60)

The cutoff brane is parameterized by

$\begin{split}-x_{m}&\approx-|y|\tanh{\kappa_{1}}-\kappa_{2}\cosh^{-2}\kappa_{1}+\mathcal{O}(\kappa_{2}^{2}),\\\
z_{m}&\approx|y|\cosh^{-1}{\kappa_{1}}-\kappa_{2}\tanh\kappa_{1}\cosh^{-1}\kappa_{1}+\mathcal{O}(\kappa_{2}^{2})\,,\end{split}$
(61)

which coincides with the JT brane (59) upon the identification,

$\kappa_{1}=\rho_{0},\qquad\kappa_{2}={\rho_{0}\bar{\phi}_{r}}=6L\mu.$ (62)

The explicit derivation of the above results is given in Appendix. A.

Again we assume entanglement islands can emerge on the left hand side $x<0$,
and calculate the entanglement entropy for intervals on the right hand side
$x>0$ via both the island formula and the RT formula. Then compare the results
with those in AdS/BCFT with fluctuating brane. Let us consider the interval
$A=[0,L]$ and assume the island region $\mathrm{Is}(A)=[-a,0)$, then the
island formula gives

$\begin{split}S([0,L])&=\min_{a}\left\\{\frac{c}{6}\log\frac{(L+a)^{2}}{2a\epsilon}+\frac{c}{6}\kappa_{1}+\frac{c}{6}\frac{\kappa_{2}}{a}\right\\}\\\
&\approx\frac{c}{6}\log\frac{2L}{\epsilon}+\frac{c}{6}\kappa_{1}+\frac{c}{6}\frac{\kappa_{2}}{L},\end{split}$
(63)

which aligns with (45) upon the identifications (62). The above formula is
minimized at

$x\approx-a_{\rm bdy}=-L-12L\mu,$ (64)

which also coincides with (44).

Holographically, $S_{A}$ is given by the RT surface connecting the two
boundary points at $x=-a_{\rm bdy},L$, which is cut off at the cutoff sphere
centered at $x=-a_{\rm bdy}$. Note that the point $x=-a_{\rm bdy}$ locates on
the asymptotic boundary, i.e. $y=-a_{\rm bdy},\rho\to\infty$, while the cutoff
point $y=-a_{\rm brane}$ is on the cutoff brane. Since $a_{\rm bdy}\neq L$,
the center of the RT curve is no longer the origin, rather it is settled at

$x=-x_{0}\approx\frac{-a_{\rm bdy}+L}{2}=-6L\mu.$ (65)

We now need to check two features for the RT curve to match it with the RT
surface anchored on the KR brane in AdS/BCFT. Firstly, the cutoff point on the
RT curve should lie on the cutoff brane. Secondly, the RT curve should be
normal to the cutoff brane at the cutoff point. If the first feature holds,
then the tangent line of the cutoff brane at the cutoff point should pass the
center of this RT curve, as shown in the right figure of Fig. 6. According to
(139), the intersection point of this tangent line and the asymptotic boundary
is given by

$\displaystyle x=-\frac{\kappa_{2}a_{\rm bdy}}{\kappa_{2}+a_{\rm
bdy}}\approx-6L\mu\,,$ (66)

which, as expected, coincides with the center of the RT circle up to order
$\mathcal{O}(\mu)$, hence the first feature is confirmed. Moreover, since the
RT curve is normal to the cutoff sphere and the cutoff brane is tangent to the
cutoff sphere, the RT curve should be normal to the cutoff brane, thus the
second feature is confirmed. Furthermore, the $y$ coordinate of the cutoff
point is given by

$y=-a_{\rm brane}\approx-L-6L(\tanh\rho_{0}+1)\mu\neq-a_{\rm bdy},$ (67)

which coincides with the point (48) at which the RT surface anchors on the KR
brane in AdS/BCFT.

The fact that $a_{\rm bdy}\neq a_{\rm brane}$ is well understood in the Weyl
transformed CFT2 set-up. The naive reason is that, due to the fluctuation of
the brane, the RT circle of the interval $[0,L]$ is no longer centered as the
origin, which is obvious in Fig.6. Nevertheless, this inequality raises up an
important physical confusion in the set-up of JT gravity on the brane coupled
to a bath CFT2. In this set-up, since the gravity part of the $2d$ effective
theory is on the KR brane, the additional twist operator due to the island
formula should be inserted at $y=-a_{\rm brane}$, which is also the point
where the RT curve anchors on the brane. This indicates $a_{\rm bdy}=a_{\rm
brane}$, which is inconsistent with our observation. As we have shown, this
insertion will not reproduce the result from the RT formula in doubly
holography. Our discussion indicates that the twist operator should be
inserted at $y=-a_{\rm bdy}$. This confusion can be fully understood in the
holographic Weyl transformed set-up, where the dual field theory is always on
the asymptotic boundary rather than the cutoff brane. This leads us to a sharp
conjecture that, all the AdS/BCFT set-ups are essentially a holographic Weyl
transformed CFT2.

##  4 Entanglement wedge cross-sections with fluctuating brane

Here we classify all possible phases for the entanglement wedge of $A\cup B$
with a fluctuating KR brane, and calculate the area of the corresponding EWCS.
We set $A=[b_{1},b_{2}]$ and $B=[b_{3},b_{4}]$ with $0\leq b_{1}<b_{2}\leq
b_{3}<b_{4}$. Adjacency between regions $A$ and $B$ happens when
$b_{2}=b_{3}$. We present illustrations of all classifications for EWCS in
Figs. 7 and 8. Compared with the configuration with non-fluctuating brane [2],
computing the area of EWCS is more complicated in the present scenario.
Fortunately, as we will show below, perturbing the results in the non-
fluctuating configurations allows us to obtain the area of EWCS up to the
first order easily.

###  4.1 No-island phase

Figure 7: EWCS for phases A1 and D1

We first consider the simplest cases, where $AB$ admits no island, hence the
EWCS coincides with those in AdS3/CFT2. We denote such no-island phases as
phase A1 and B1 (see in Fig. 7), where $A$ and $B$ are adjacent and disjoint
respectively. The corresponding area of the EWCSs are those given in [44]

$\begin{split}&\textbf{Phase-A1
:}\frac{\operatorname{Area}\left[\Sigma_{AB}\right]}{4G_{N}}=\frac{c}{6}\log\frac{2\left(b_{2}-b_{1}\right)\left(b_{4}-b_{2}\right)}{\epsilon\left(b_{4}-b_{1}\right)},\\\
&\textbf{Phase-D1
:}\frac{\operatorname{Area}\left[\Sigma_{AB}\right]}{4G_{N}}=\frac{c}{6}\log\frac{1+\sqrt{w}}{1-\sqrt{w}},\quad
w=\frac{\left(b_{4}-b_{3}\right)\left(b_{2}-b_{1}\right)}{\left(b_{3}-b_{1}\right)\left(b_{4}-b_{2}\right)}.\end{split}$
(68)

###  4.2 Island phases

Now we consider the cases where $AB$ admits an island. When $A$ and $B$ are
adjacent, we denote the phases as A2, otherwise we denote them as D2.
According to the position where the EWCS anchors in the bulk, phases A2 and D2
can further be decomposed into the following three classes:

  * •

Phases A2a and D2a: the EWCS is anchored on the RT($b_{1}$),

  * •

Phases A2b and D2b: the EWCS is anchored on the brane,

  * •

Phases A2c and D2c: the EWCS is anchored on the RT($b_{4}$),

where RT($b_{i}$) denotes the RT surface that emanates from the boundary point
$x=b_{i}$ and anchors on the brane (see Fig. 8).

As was pointed out in Sec. 3.3, compared with the non-fluctuating
configurations, the center of the RT($b_{i}$) circle is no longer the origin
$x=0$. Rather, it is settled at (65)

$\displaystyle x=-x_{0}=-6b_{i}\mu=-\rho_{0}\bar{\phi}_{r}\,.$ (69)

It is important to note that, $x_{0}$ is independent of $b_{i}$, hence the
centers of all RT$(b_{i})$ circles are the same. For convenience, later we
will expand our results with respect to $x_{0}$.

Figure 8: EWCS for phases A2 and D2

In phase A2a, the EWCS emanates from the boundary point $x=b_{2}$ and anchors
on RT($b_{1}$) vertically. Since the center of RT($b_{i}$) settles at
$x=-x_{0}$, the whole configuration differs from the non-fluctuating one by a
simple translation $x\to x+x_{0}$. Therefore, the area of the EWCS can be
easily obtained from the results in the non-fluctuating configuration, with
the replacement $b_{1}\to b_{1}+x_{0}$ and $b_{2}\to b_{2}+x_{0}$,

$\displaystyle\frac{\text{Area}(\Sigma_{AB})}{4G_{N}}=$
$\displaystyle\frac{c}{6}\log\frac{b_{2}^{2}-b_{1}^{2}}{b_{1}\epsilon}\Big{|}_{b_{1}\to
b_{1}+x_{0},~{}b_{2}\to b_{2}+x_{0}}$ (70) $\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{(b_{2}+x_{0})^{2}-(b_{1}+x_{0})^{2}}{(b_{1}+x_{0})\epsilon}$
(71) $\displaystyle\approx$
$\displaystyle\frac{c}{6}\log\frac{b_{2}^{2}-b_{1}^{2}}{b_{1}\epsilon}-\frac{c}{6}\frac{b_{2}-b_{1}}{b_{1}+b_{2}}\frac{x_{0}}{b_{1}}.$
(72)

One can extend the EWCS to the asymptotic boundary and find intersection point
is given by

$\tilde{q}_{1}=\frac{(b_{1}+x_{0})^{2}}{b_{2}+x_{0}}-x_{0}\approx\frac{b_{1}^{2}}{b_{2}}-\frac{(b_{1}-b_{2})^{2}}{b_{2}^{2}}x_{0}.$
(73)

We will see that, this intersection point coincides with the balance point,
which solves the balance requirements.

In phase A2b, the EWCS is just the RT($b_{2}$) circle, whose area is given by
(49) with the replacement $L\to b_{2}$,

$\displaystyle\frac{\text{Area}(\Sigma_{AB})}{4G_{N}}$
$\displaystyle=\frac{c}{6}\log\frac{2b_{2}}{\epsilon}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{b_{2}}.$
(74)

In phase A2c, the EWCS in phase A2c anchors on the RT($b_{4}$) circle, and its
area can be calculated in similar way, which is just the result in the non-
fluctuating configurations with a translation,

$\begin{split}\frac{\text{Area}(\Sigma_{AB})}{4G_{N}}&=\frac{c}{6}\log\frac{b_{4}^{2}-b_{2}^{2}}{b_{4}\epsilon}\Big{|}_{b_{2}\to
b_{2}+x_{0},~{}b_{4}\to
b_{4}+x_{0}}\cr&\approx\frac{c}{6}\log\frac{b_{4}^{2}-b_{2}^{2}}{b_{4}\epsilon}-\frac{c}{6}\frac{b_{4}-b_{2}}{b_{4}+b_{2}}\frac{x_{0}}{b_{4}}.\end{split}$
(75)

Extending the EWCS to the asymptotic boundary, we get

$\tilde{q}_{1}\approx\frac{b_{4}^{2}}{b_{2}}-\frac{(b_{4}-b_{2})^{2}}{b_{2}^{2}}x_{0}.$
(76)

In phase D2a, the EWCS anchors on the RT($b_{1}$) circle and the RT surface
for the interval $[b_{2},b_{3}]$. The area of the EWCS is given by

$\begin{split}\frac{\text{Area}(\Sigma_{AB})}{4G_{N}}=&\frac{c}{6}\log\frac{b_{2}b_{3}-b_{1}^{2}+\sqrt{(b_{3}^{2}-b_{1}^{2})(b_{2}^{2}-b_{1}^{2})}}{b_{1}\left(b_{3}-b_{2}\right)}\Big{|}_{b_{1}\to
b_{1}+x_{0},~{}b_{2}\to b_{2}+x_{0}}\\\
=&\frac{c}{6}\log\left(\frac{b_{2}b_{3}-b_{1}^{2}+\sqrt{\left(b_{2}^{2}-b_{1}^{2}\right)\left(b_{3}^{2}-b_{1}^{2}\right)}}{b_{1}\left(b_{3}-b_{2}\right)}\right)-\frac{c}{6}\frac{\sqrt{\left(b_{2}^{2}-b_{1}^{2}\right)\left(b_{3}^{2}-b_{1}^{2}\right)}}{b_{1}(b_{1}+b_{2})(b_{1}+b_{3})}x_{0}.\end{split}$
(77)

Extend the EWCS to the asymptotic boundary, and we get

$\begin{split}\tilde{q}_{1}\approx&\frac{b_{1}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{1}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{1}(b_{1}+b_{2})(b_{1}+b_{3})+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}(2b_{1}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{1}+b_{2})(b_{1}+b_{3})}\right)x_{0},\end{split}$
(78)
$\begin{split}\tilde{q}_{2}\approx&\frac{b_{1}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{1}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{1}(b_{1}+b_{2})(b_{1}+b_{3})-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}(2b_{1}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{1}+b_{2})(b_{1}+b_{3})}\right)x_{0}.\end{split}$
(79)

In phase D2b, the EWCS anchors on the brane and the RT surface for the
interval $[b_{2},b_{3}]$. The derivation for the area of EWCS gets tricky. We
leave it in Appendix. B. The results are as follows. The area of EWCS is

$\begin{split}\frac{\text{Area}(\Sigma_{AB})}{4G_{N}}=\frac{c}{6}\log\frac{\sqrt{b_{3}}+\sqrt{b_{2}}}{\sqrt{b_{3}}-\sqrt{b_{2}}}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{\sqrt{b_{2}b_{3}}}.\end{split}$
(80)

And the intersection point with the asymptotic boundary of AdS is

$\tilde{q}_{2}=\sqrt{b_{2}b_{3}}+3(\sqrt{b_{3}}-\sqrt{b_{2}})^{2}\frac{x_{0}}{6\sqrt{b_{2}b_{3}}}.$
(81)

In phase D2c, the EWCS anchors on RT$(b_{4})$. Similar to phase D2c, the area
of the EWCS for phase D2c can be easily obtained from the non-fluctuating case
with a translation,

$\begin{split}\frac{\text{Area}(\Sigma_{AB})}{4G_{N}}=&\frac{c}{6}\log\left(\frac{b_{2}b_{3}-b_{4}^{2}+\sqrt{\left(b_{2}^{2}-b_{4}^{2}\right)\left(b_{3}^{2}-b_{4}^{2}\right)}}{b_{4}\left(b_{3}-b_{2}\right)}\right)\Big{|}_{b_{4}\to
b_{4}+x_{0},~{}b_{2}\to b_{2}+x_{0}}\\\
=&\frac{c}{6}\log\left(\frac{b_{2}b_{3}-b_{4}^{2}+\sqrt{\left(b_{2}^{2}-b_{4}^{2}\right)\left(b_{3}^{2}-b_{4}^{2}\right)}}{b_{4}\left(b_{3}-b_{2}\right)}\right)-\frac{c}{6}\frac{\sqrt{\left(b_{2}^{2}-b_{4}^{2}\right)\left(b_{3}^{2}-b_{4}^{2}\right)}}{b_{4}(b_{4}+b_{2})(b_{4}+b_{3})}x_{0}.\end{split}$
(82)

Two intersection points are

$\begin{split}\tilde{q}_{1}&\approx\frac{b_{4}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{4}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{4}(b_{4}+b_{2})(b_{4}+b_{3})-\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}(2b_{4}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{4}+b_{2})(b_{4}+b_{3})}\right)x_{0},\end{split}$
(83)
$\begin{split}\tilde{q}_{2}&\approx\frac{b_{4}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{4}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{4}(b_{4}+b_{2})(b_{4}+b_{3})+\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}(2b_{4}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{4}+b_{2})(b_{4}+b_{3})}\right)x_{0}.\end{split}$
(84)

##  5 BPE in Weyl transformed CFT

In this section, we study the BPE in the AdS/BCFT set-up with fluctuating
brane, and match it with the area of the EWCS. The computation of the BPE in
the fluctuating brane configurations is similar to that in non-fluctuating
configurations [2], and our foundation can be summarized into the following
three proposals:

  1. 1.

As in the no-island phases, the PEE structure of the island phase is also
described by the two-point PEEs $\mathcal{I}(x,y)$.

  2. 2.

When computing the PEE between any two spacelike separated regions, we should
also properly take into account the contributions from their island regions as
well as from the ownerless islands.

  3. 3.

The two-point functions between the endpoints of any interval $[a,b]$ in the
Weyl transformed CFT are well defined and given by (53). Furthermore, this
two-point function represents the PEE following the basic proposal (22).

Compared with the non-fluctuating configurations [2], the position where the
EWCS anchors on the brane and the position of the balance point on the
asymptotic boundary differ from each other, which is akin to the observation
that $a_{\rm bdy}\neq a_{\rm brane}$ in our discussion on the entanglement
entropy.

The simplest cases are the phases A1 and D1, where $AB$ admits no island and
the correlations between $A$ and $B$ receive no contribution from the islands.
In these cases, the PEE, BPE and EWCS coincide with their counterpart in
AdS3/CFT2, see Fig.7. In the following, we focus on the cases where $AB$
admits a island.

###  5.1 Adjacent $AB$ with island

Let us first consider the adjacent case, where the island region
$\mathrm{Is}(AB)$ in the Weyl transformed CFT is given by
$\text{Is}(AB)=[-a_{4},-a_{1}]$ with $a_{i}\approx b_{i}+2x_{0}\,,~{}i=1,4$.
And the cross-section has three different saddle points, which are shown in
Figs. 9, 10 and 11 respectively. The assignment for the ownerless island
region also has three possibilities:

  1. 1.

$\operatorname{Ir}(A)=\emptyset,\quad\operatorname{Ir}(B)=[-a_{4},-a_{1}]$.

  2. 2.

$\operatorname{Ir}(A)=[-q,-a_{1}],\quad\operatorname{Ir}(B)=[-a_{4},-q]$ with
$a_{1}<q<a_{4}$.

  3. 3.

$\operatorname{Ir}(A)=[-a_{4},-a_{1}],\quad\operatorname{Ir}(B)=\emptyset$.

Following the discussions in [2], for the above assignments 1, 2, and 3, we
can solve the balance requirements and get a corresponding BPE, which exactly
matches the three saddles of the cross-section shown in Figs. 9, 10 and 11. We
ought to select the assignment that yields the minimal BPE, corresponding to
the area of minimal cross-section, i.e. the EWCS.

Figure 9: Phase A2a.

In phase A2a, the EWCS anchors on RT($b_{1})$ and we should consider the
assignment $\operatorname{Ir}(A)=\emptyset$,
$\operatorname{Ir}(B)=[-a_{4},-a_{1}]$. Let us assume

$\quad\text{Ir}(A_{1})=\emptyset,\quad\text{Ir}(B_{1})=(-\infty,-a_{4}]\cup[-a_{1},0].$
(85)

In this case the partition point satisfies $0<q_{1}<b_{1}$, see Fig. 9.

Now we solve the balanced requirements. Using the generalized ALC formula
(24), we get the two relevant PEEs

$\displaystyle\mathcal{I}\left(A,B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)\right)$
$\displaystyle=\frac{1}{2}\left[\tilde{S}_{AA_{1}}+\tilde{S}_{A}-\tilde{S}_{A_{1}}\right]$
(86)
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{\left[q_{1},b_{2}\right]}+\tilde{S}_{\left[b_{1},b_{2}\right]}-\tilde{S}_{\left[q_{1},b_{1}\right]}\right)$
$\displaystyle=\frac{c}{6}\log\frac{(b_{2}-q_{1})(b_{2}-b_{1})}{(b_{1}-q_{1})\epsilon},$
$\displaystyle\mathcal{I}\left(B\operatorname{Ir}(B),AA_{1}\right)$
$\displaystyle=\frac{1}{2}\left[\tilde{S}_{B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)}+\tilde{S}_{B\operatorname{Ir}(B)}-\tilde{S}_{B_{1}\operatorname{Ir}\left(B_{1}\right)}\right]$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{\left[q_{1},b_{2}\right]}+\tilde{S}_{\left[-b_{4},-b_{1}\right]\cup\left[b_{2},b_{4}\right]}-\tilde{S}_{\left[-\infty,-b_{4}\right]\cup\left[-b_{1},q_{1}\right]\cup\left[b_{4},\infty\right]}\right)$
$\displaystyle=\frac{c}{6}\log\frac{\left(b_{2}-q_{1}\right)\left(b_{2}+a_{1}\right)}{\left(q_{1}+a_{1}\right)\epsilon}.$

Then solving the balance condition
$\mathcal{I}\left(A,B\operatorname{Ir}(B)\cup
B_{1}\operatorname{Ir}\left(B_{1}\right)\right)=\mathcal{I}\left(B\operatorname{Ir}(B),AA_{1}\right)$,
we obtain the balance point

$q_{1}\approx\frac{b_{1}^{2}}{b_{2}}-\frac{(b_{1}-b_{2})^{2}}{b_{2}^{2}}x_{0}.$
(87)

Furthermore, we can get the following BPE,

$\begin{split}\text{BPE}(A:B)=&\frac{c}{6}\log\frac{(b_{2}-q_{1})}{\epsilon}\frac{(b_{2}-b_{1})}{(b_{1}-q_{1})}\approx\frac{c}{6}\log\frac{b_{2}^{2}-b_{1}^{2}}{b_{1}\epsilon}-\frac{c}{6}\frac{b_{2}-b_{1}}{b_{1}+b_{2}}\frac{x_{0}}{b_{1}},\end{split}$
(88)

which matches the area of EWCS (70) exactly.

One can also assume that $\text{Ir}(A_{1})\neq\emptyset$. Nevertheless,
similar to the non-fluctuating configurations studied in [2], the solution to
the balance requirements does not exist under such an assumption.

Figure 10: Phase A2b

In Phase A2b, the EWCS anchors on the brane and we should consider
$\operatorname{Ir}(A)=[-q,-a_{1}]$, and $\operatorname{Ir}(B)=[-a_{4},-q]$.
Since $A$ and $B$ in this phase are symmetric in the sense that they both
receive island contributions, it is natural to assume both $A_{1}$ and $B_{1}$
admit island contributions $\text{Ir}(A_{1})$ and $\text{Ir}(B_{1})$. In
particular, we choose

$A_{1}=[0,a_{1}],\quad B_{1}=[a_{4},+\infty),$ (89)

and

$\text{Ir}(A_{1})=[-a_{1},0],\quad\text{Ir}(B_{1})=(-\infty,-a_{4}].$ (90)

Then using the generalized ALC formula (24), we get

$\displaystyle\mathcal{I}\left(A\operatorname{Ir}(A),B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)\right)$
$\displaystyle=\frac{1}{2}\left[\tilde{S}_{A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)}+\tilde{S}_{A\operatorname{Ir}(A)}-\tilde{S}_{A_{1}\operatorname{Ir}\left(A_{1}\right)}\right]$
(91)
$\displaystyle=\frac{1}{2}\left[\tilde{S}_{\left[-q,b_{2}\right]}+\tilde{S}_{\left[-q,-b_{1}\right]\cup\left[b_{1},b_{2}\right]}-\tilde{S}_{\left[-b_{1},b_{1}\right]}\right]$
$\displaystyle=\tilde{S}_{\left[-q,b_{2}\right]},$
$\displaystyle\mathcal{I}\left(B\operatorname{Ir}(B),A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)\right)$
$\displaystyle=\frac{1}{2}\left[\tilde{S}_{B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)}+\tilde{S}_{B\operatorname{Ir}(B)}-\tilde{S}_{B_{1}\operatorname{Ir}\left(B_{1}\right)}\right]$
(92)
$\displaystyle=\frac{1}{2}\left[\tilde{S}_{\left[-q,b_{2}\right]}+\tilde{S}_{\left[-b_{4},-q\right]\cup\left[b_{2},b_{4}\right]}-\tilde{S}_{\left[-b_{4},b_{4}\right]}\right]$
$\displaystyle=\tilde{S}_{\left[-q,b_{2}\right]},$

with

$\tilde{S}_{\left[-q,b_{2}\right]}=\frac{c}{6}\log\frac{(q+b_{2})^{2}}{2q\epsilon}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{q}.$
(93)

Thus the balanced requirement is automatically satisfied regardless of the
choice of $q\in[-a_{4},-a_{1}]$. However, different $q$ will give different
BPEs, and according to the minimal requirement, the minimal one is taken at

$\frac{\mathrm{d}}{\mathrm{d}q}\mathcal{I}\left(A\operatorname{Ir}(A),B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)\right)=0\quad\Rightarrow\quad
q=b_{2}+2x_{0}.$ (94)

Inserting (94) into (93), we get the BPE

$\text{BPE}(A:B)=\frac{c}{6}\rho_{0}+\frac{c}{6}\log\frac{2b_{2}}{\epsilon}+\frac{c}{6}\frac{x_{0}}{b_{2}},$
(95)

which is exactly the area of EWCS (74).

In Phase A2c, the EWCS anchors on RT$(b_{4})$ and we should consider
$\operatorname{Ir}(A)=[-a_{4},-a_{1}]$ and $\operatorname{Ir}(B)=\emptyset$.
Following the similar argument as phase A2a, we arrive at

$q_{1}\approx\frac{b_{4}^{2}}{b_{2}}-\frac{(b_{4}-b_{2})^{2}}{b_{2}^{2}}x_{0},$
(96)
$\begin{split}\text{BPE}(A:B)\approx&\frac{c}{6}\log\frac{b_{4}^{2}-b_{2}^{2}}{b_{4}\epsilon}-\frac{c}{6}\frac{b_{4}-b_{2}}{b_{2}+b_{4}}\frac{x_{0}}{b_{4}},\end{split}$
(97)

which coincides with the area of the EWCS (75).

Figure 11: Phase A2c

### Minimizing BPE for phase A2

The last step of the generalized ALC proposal is to choose the minimal BPE.
Now let us compare the different BPEs in phase A2.

The critical point between phase A2a and A2b is given by

$f(b_{1},b_{2})\equiv\rho_{0}+\log\frac{2b_{1}b_{2}}{(b_{2}^{2}-b_{1}^{2})\epsilon}+\frac{(b_{1}^{2}+b_{2}^{2})x_{0}}{b_{1}b_{2}(b_{1}+b_{2})}=0.$
(98)

Phase A2a has the smaller BPE when $f(b_{4},b_{2})>0$. Note that we assume
$A_{1}A$ admits no island in phase A2a. Now we show that this assumption is
consistent with the minimal condition $f(b_{4},b_{2})>0$ for phase A2a. Let us
compare the island and non-island saddles of the entanglement entropy for
$A_{1}A$

$\displaystyle S_{\mathrm{island}}\left(A_{1}A\right)-S_{\mathrm{non-
island}}\left(A_{1}A\right)$ (99) $\displaystyle=$
$\displaystyle\frac{c}{3}\rho_{0}+\frac{c}{6}\log\frac{4q_{1}b_{2}}{\epsilon^{2}}+\frac{c}{6}\left(\frac{1}{b_{2}}+\frac{1}{q_{1}}\right)x_{0}-\frac{c}{6}\log\frac{(b_{2}-q_{1})^{2}}{\epsilon^{2}}$
$\displaystyle=$ $\displaystyle\frac{c}{3}f(b_{1},b_{2})>0,$

where we have used the balance point (87). Thus when phase A2a takes the
smaller BPE, $A_{1}A$ admits no island.

Similarly, the critical point between phase A2c and A2b is given by

$f(b_{4},b_{2})\equiv\rho_{0}+\log\frac{2b_{4}b_{2}}{(b_{4}^{2}-b_{2}^{2})\epsilon}+\frac{(b_{2}^{2}+b_{4}^{2})x_{0}}{b_{2}b_{4}(b_{2}+b_{4})}=0.$
(100)

And phase A2c has the smaller BPE when $f(b_{4},b_{2})>0$. Follow the similar
argument, one could show that $BB_{1}$ admits no island in phase A2c under
$f(b_{4},b_{2})>0$.

###  5.2 Disjoint $AB$ with island

Now we consider the disjoint case where the island region of $AB$ in Weyl
transformed CFT is given by $\text{Is}(AB)=[-a_{4},-a_{1}]$ where
$a_{i}\approx b_{i}+2x_{0}$ with $i=1,4$. And the three different assignments
of the ownerless island are

  1. 1:

$\operatorname{Ir}(A)=\emptyset,\quad\operatorname{Ir}(B)=[-a_{4},-a_{1}]$.

  2. 2:

$\operatorname{Ir}(A)=[-q,-a_{1}],\quad\operatorname{Ir}(B)=[-a_{4},-q]$ with
$a_{1}<q<a_{4}$.

  3. 3:

$\operatorname{Ir}(A)=[-a_{4},-a_{1}],\quad\operatorname{Ir}(B)=\emptyset$.

Akin to adjacent case, the above assignments lead to three distinct results
for BPE, which match the areas of three saddles of the cross-section shown in
Fig. 12, 13 and 14, respectively.

Figure 12: Phase D2a

In Phase D2a, the EWCS anchors on the RT($b_{1}$) and the RT surface of the
interval $[b_{2},b_{3}]$ and we should consider the assignment
$\operatorname{Ir}(A)=\emptyset,\ \operatorname{Ir}(B)=[-a_{4},-a_{1}]$. Let
us assume

$0<q_{1}<b_{1},\quad\text{Ir}(A_{1})=\emptyset,\quad\text{Ir}(B_{1})=(-\infty,-a_{4}]\cup[-a_{1},0].$
(101)

Now we solve the two balance conditions. By the generalized ALC formula (25),
the four PEEs are given by

$\displaystyle\mathcal{I}\left(A,B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)B_{2}\right)=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{AA_{1}}+\tilde{S}_{A_{2}}-\tilde{S}_{A_{1}}-\tilde{S}_{A_{2}}\right]$
(102) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{\left[q_{1},b_{2}\right]}+\tilde{S}_{\left[b_{1},q_{2}\right]}-\tilde{S}_{\left[q_{1},b_{1}\right]}-\tilde{S}_{\left[b_{2},q_{2}\right]}\right]$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{b_{2}-q_{1}}{q_{2}-b_{2}}\frac{q_{2}-b_{1}}{b_{1}-q_{1}},$
$\displaystyle\mathcal{I}\left(B\operatorname{Ir}(B),A_{1}A_{2}A\right)=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)}+\tilde{S}_{B\operatorname{Ir}(B)B_{2}}-\tilde{S}_{B_{2}}-\tilde{S}_{B_{1}\operatorname{Ir}\left(B_{1}\right)}\right]$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{b_{3}-q_{1}}{b_{3}-q_{2}}\frac{q_{2}+a_{1}}{q_{1}+a_{1}},$

and

$\displaystyle\mathcal{I}\left(B_{2},A_{1}A_{2}A\right)=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)B_{2}}+\tilde{S}_{B_{2}}-\tilde{S}_{B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)}\right]$
(103) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{\left[q_{1},q_{2}\right]}+\tilde{S}_{\left[q_{2},b_{3}\right]}-\tilde{S}_{\left[q_{1},b_{3}\right]}\right]$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{\left(q_{2}-q_{1}\right)\left(b_{3}-q_{2}\right)}{\epsilon\left(b_{3}-q_{1}\right)},$
$\displaystyle\mathcal{I}\left(A_{2},B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)B_{2}\right)=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{AA_{1}A_{2}}+\tilde{S}_{A_{2}}-\tilde{S}_{AA_{1}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{\left[q_{1},q_{2}\right]}+\tilde{S}_{\left[b_{2},q_{2}\right]}-\tilde{S}_{\left[q_{1},b_{2}\right]}\right]$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{\left(q_{2}-q_{1}\right)\left(q_{2}-b_{2}\right)}{\left(b_{2}-q_{1}\right)\epsilon}.$

Solving the balanced conditions, we get the balanced points

$\begin{split}q_{1}&\approx\frac{b_{1}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{1}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{1}(b_{1}+b_{2})(b_{1}+b_{3})+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}(2b_{1}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{1}+b_{2})(b_{1}+b_{3})}\right)x_{0},\end{split}$
(104)
$\begin{split}q_{2}&\approx\frac{b_{1}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{1}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{1}(b_{1}+b_{2})(b_{1}+b_{3})-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}(2b_{1}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{1}+b_{2})(b_{1}+b_{3})}\right)x_{0},\end{split}$
(105)

which align with the two intersection points (78) and (79). Then the BPE is
obtained as

$\begin{split}\text{BPE}&=\frac{c}{6}\log\frac{b_{2}-q_{1}}{q_{2}-b_{2}}\frac{q_{2}-b_{1}}{b_{1}-q_{1}}\\\
&\approx\frac{c}{6}\log\left(\frac{b_{2}b_{3}-b_{1}^{2}+\sqrt{\left(b_{2}^{2}-b_{1}^{2}\right)\left(b_{3}^{2}-b_{1}^{2}\right)}}{b_{1}\left(b_{3}-b_{2}\right)}\right)-\frac{c}{6}\frac{\sqrt{\left(b_{2}^{2}-b_{1}^{2}\right)\left(b_{3}^{2}-b_{1}^{2}\right)}}{b_{1}(b_{1}+b_{2})(b_{1}+b_{3})}x_{0},\end{split}$
(106)

which matches the area of EWCS (77).

Figure 13: Phase D2b

In Phase D2b, the EWCS anchors on the brane and the RT surface of the interval
$[b_{2},b_{3}]$ and we should consider the assignment
$\operatorname{Ir}(A)=[-q,-a_{1}],\ \operatorname{Ir}(B)=[-a_{4},-q]$. Assume
that both $A_{1}$ and $B_{1}$ admit island contributions $\text{Ir}(A_{1})$
and $\text{Ir}(B_{1})$ and set

$A_{1}=[0,a_{1}],\quad B_{1}=[a_{4},+\infty),$ (107)

and accordingly

$\text{Ir}(A_{1})=[-a_{1},0],\quad\text{Ir}(B_{1})=(-\infty,-a_{4}].$ (108)

By the generalized ALC formula (25), the four PEEs are given by

$\displaystyle\mathcal{I}\left(B\operatorname{Ir}(B),A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)A_{2}\right)$
(109) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)}+\tilde{S}_{B_{2}B\operatorname{Ir}(B)}-\tilde{S}_{B_{2}}-\tilde{S}_{B_{1}\operatorname{Ir}\left(B_{1}\right)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{S}_{[-\infty,-q]\cup\left[b_{3},+\infty\right]}+\tilde{S}_{\left[-b_{4},-q\right]\cup\left[q_{2},b_{4}\right]}-\tilde{S}_{\left[q_{2},b_{3}\right]}-\tilde{S}_{\left[-\infty,-b_{4}\right]\cup\left[b_{4},+\infty\right]}\right)$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{\left(b_{3}+q\right)\left(q+q_{2}\right)}{2q\left(b_{3}-q_{2}\right)}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{q},$
$\displaystyle\mathcal{I}\left(A\operatorname{Ir}(A),B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)B_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)}+\tilde{S}_{A_{2}A\operatorname{Ir}(A)}-\tilde{S}_{A_{1}\operatorname{Ir}\left(A_{1}\right)}-\tilde{S}_{A_{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{S}_{\left[-q,b_{2}\right]}+\tilde{S}_{\left[-q,-b_{1}\right]\cup\left[b_{1},q_{2}\right]}-\tilde{S}_{\left[-b_{1},b_{1}\right]}-\tilde{S}_{\left[b_{2},q_{2}\right]}\right)$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{\left(b_{2}+q\right)\left(q_{2}+q\right)}{2q\left(q_{2}-b_{2}\right)}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{q},$

and

$\displaystyle\mathcal{I}\left(B_{2},A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)A_{2}\right)$
(110) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{BB_{1}\operatorname{Ir}\left(BB_{1}\right)B_{2}}+\tilde{S}_{B_{2}}-\tilde{S}_{BB_{1}\operatorname{Ir}\left(BB_{1}\right)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{S}_{[-\infty,-q]\cup\left[q_{2},\infty\right]}+\tilde{S}_{\left[q_{2},b_{3}\right]}-\tilde{S}_{[-\infty,-q]\cup\left[b_{3},+\infty\right]}\right)$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{\left(q_{2}+q\right)\left(b_{3}-q_{2}\right)}{\epsilon\left(b_{3}+q\right)},$
$\displaystyle\mathcal{I}\left(A_{2},B\operatorname{Ir}(B)B_{1}\operatorname{Ir}\left(B_{1}\right)B_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\tilde{S}_{A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)A_{2}}+\tilde{S}_{A_{2}}-\tilde{S}_{A\operatorname{Ir}(A)A_{1}\operatorname{Ir}\left(A_{1}\right)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{S}_{\left[-q,q_{2}\right]}+\tilde{S}_{\left[b_{2},q_{2}\right]}-\tilde{S}_{\left[-q,b_{2}\right]}\right)$
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\frac{\left(q_{2}+q\right)\left(q_{2}-b_{2}\right)}{\epsilon\left(b_{2}+q\right)}.$

We find that the two balance conditions are degenerate:

$\frac{b_{3}+q}{b_{3}-q_{2}}=\frac{b_{2}+q}{q_{2}-b_{2}}\quad\Rightarrow\quad
q_{2}=\frac{2b_{2}b_{3}+(b_{2}+b_{3})q}{b_{2}+b_{3}+2q}.$ (111)

Like the phase A2b, the BPE changes with $q$. By varying $q$, we find the
minimal BPE

$\begin{split}\text{BPE}(A:B)=\frac{c}{6}\log\frac{\sqrt{b_{3}}+\sqrt{b_{2}}}{\sqrt{b_{3}}-\sqrt{b_{2}}}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{\sqrt{b_{2}b_{3}}},\end{split}$
(112)

which matches the area of EWCS (80). The extremal point is given by

$q\approx\sqrt{b_{2}b_{3}}+3(\sqrt{b_{2}}+\sqrt{b_{3}})^{2}\frac{x_{0}}{6\sqrt{b_{2}b_{3}}},$
(113)

Inserting (113) to (111), we get the corresponding balance point

$q_{2}\approx\sqrt{b_{2}b_{3}}+3(\sqrt{b_{3}}-\sqrt{b_{2}})^{2}\frac{x_{0}}{6\sqrt{b_{2}b_{3}}},$
(114)

which is exactly the intersection point (81) by extending the EWCS to the
asymptotic boundary.

Figure 14: Phase D2c

In Phase D2c, the EWCS anchors on the RT($b_{4}$) and the RT surface of the
interval $[b_{2},b_{3}]$ and we should consider the assignment
$\operatorname{Ir}(B)=\emptyset,\ \operatorname{Ir}(A)=[-a_{4},-a_{1}]$.
Following the similar procedure as in phase D2a, we get two balance points,

$\begin{split}q_{1}&\approx\frac{b_{4}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{4}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{4}(b_{4}+b_{2})(b_{4}+b_{3})-\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}(2b_{4}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{4}+b_{2})(b_{4}+b_{3})}\right)x_{0},\end{split}$
(115)
$\begin{split}q_{2}&\approx\frac{b_{4}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{4}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{4}(b_{4}+b_{2})(b_{4}+b_{3})+\sqrt{(b_{2}^{2}-b_{4}^{2})(b_{3}^{2}-b_{4}^{2})}(2b_{4}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{4}+b_{2})(b_{4}+b_{3})}\right)x_{0},\end{split}$
(116)

which are exactly the intersection points (83) and (84). The corresponding BPE
is

$\displaystyle\text{BPE}(A:B)$
$\displaystyle=\frac{c}{6}\log\left(\frac{b_{2}b_{3}-b_{4}^{2}+\sqrt{\left(b_{2}^{2}-b_{4}^{2}\right)\left(b_{3}^{2}-b_{4}^{2}\right)}}{b_{4}\left(b_{3}-b_{2}\right)}\right)-\frac{c}{6}\frac{\sqrt{\left(b_{2}^{2}-b_{4}^{2}\right)\left(b_{3}^{2}-b_{4}^{2}\right)}}{b_{4}(b_{4}+b_{2})(b_{4}+b_{3})}x_{0},$
(117)

which coincides with the area of EWCS (82).

### Minimizing BPE for phase D2

Now let us compare the BPEs in phase D2.

The critical point between phase D2a and D2b is

$\begin{split}F(b_{1},b_{2},b_{3})=0,\end{split}$ (118)

where

$F(b_{1},b_{2},b_{3})\equiv\log\frac{b_{3}b_{2}-b_{1}^{2}+\sqrt{\left(b_{3}^{2}-b_{1}^{2}\right)\left(b_{2}^{2}-b_{1}^{2}\right)}}{b_{1}\left(\sqrt{b_{3}}+\sqrt{b_{2}}\right)^{2}}-\frac{x_{0}}{\sqrt{b_{2}b_{3}}}-\frac{x_{0}\sqrt{\left(b_{2}^{2}-b_{1}^{2}\right)\left(b_{3}^{2}-b_{1}^{2}\right)}}{b_{1}(b_{1}+b_{2})(b_{1}+b_{3})}-\rho_{0}.$
(119)

When $F<0$, phase D2a dominates and gives the smaller BPE. In fact, $F<0$ is
also the condition that $A_{1}AA_{2}$ admits no island. Now we compare the
entanglement entropy for $A_{1}AA_{2}$ in no-island and island saddles,

$\displaystyle\frac{3}{c}(S_{\mathrm{non-
island}}\left(A_{1}AA_{2}\right)-S_{\mathrm{island}}\left(A_{1}AA_{2}\right))$
(120) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\frac{\left(q_{2}-q_{1}\right)^{2}}{4q_{1}q_{2}}-\frac{1}{2}\left(\frac{1}{q_{2}}+\frac{1}{q_{1}}\right)x_{0}-\rho_{0}$
$\displaystyle=$
$\displaystyle\log\frac{\sqrt{\left(b_{2}^{2}-b_{1}^{2}\right)\left(b_{3}^{2}-b_{1}^{2}\right)}}{b_{1}\left(b_{2}+b_{3}\right)}-\frac{(2b_{1}+b_{2}+b_{3})(b_{1}^{2}+b_{2}b_{3})x_{0}}{b_{1}(b_{1}+b_{2})(b_{1}+b_{3})(b_{3}+b_{2})}-\rho_{0}$
$\displaystyle\equiv$ $\displaystyle G(b_{1},b_{2},b_{3}).$

We observe that $G(b_{1},b_{2},b_{3})-F(b_{1},b_{2},b_{3})$ increases as
$b_{2}\to b_{3}$, which indicates that

$G(b_{1},b_{2},b_{3})-F(b_{1},b_{2},b_{3})<G(b_{1},b_{3},b_{3})-F(b_{1},b_{3},b_{3})=0.$
(121)

Inserting (121) to (120), we arrive at

$\frac{3}{c}(S_{\mathrm{non-
island}}\left(A_{1}AA_{2}\right)-S_{\mathrm{island}}\left(A_{1}AA_{2}\right))<F(b_{1},b_{2},b_{3})<0.$
(122)

This is consistent with our assumption that $A_{1}AA_{2}$ admits no island.

Likewise, for the BPEs between phase-D2b and phase-D2c, we conclude that
phase-D2c gives the smaller BPE and $B_{1}BB_{2}$ admits no island when
$F(b_{4},b_{2},b_{3})<0$.

###  5.3 BPE from minimizing the crossing PEE

The balance point was shown to be the minimized point of the crossing PEE in
vacuum CFT2 [56], the covariant CFT2 scenarios with or without gravitational
anomaly [55] and the island phase with the unfluctuating brane [2]. Moreover,
the minimized crossing PEE for adjacent phase in CFT2 is shown to be universal
and exactly produce the Markov gap [56]. In fact, the minimization condition
of the crossing PEE seems rather essential than the balanced requirement, as
it may be even well-defined in more general scenarios like the discrete
system.

In this subsection, we would like to see if the minimized point is still the
balanced point when the brane gets perturbed. Without loss of generality, we
will take phase A2a and phase D2a as examples.

For phase A2a, two crossing PEEs are

$\displaystyle\mathcal{I}\left(A,B_{1}\operatorname{Ir}\left(B_{1}\right)\right)$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{AA_{1}}+\tilde{S}_{AB\operatorname{Ir}(B)}-\tilde{S}_{A_{1}}-\tilde{S}_{B\operatorname{Ir}(B)}\right)$
(123)
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{\left[q_{1},b_{2}\right]}+\tilde{S}_{\left[-a_{4},-a_{1}\right]\cup\left[b_{1},b_{4}\right]}-\tilde{S}_{\left[q_{1},b_{1}\right]}-\tilde{S}_{\left[-a_{4},-a_{1}\right]\cup\left[b_{2},b_{4}\right]}\right)$
$\displaystyle=\frac{c}{6}\log\left[\frac{(b_{1}+a_{1})\left(b_{2}-q_{1}\right)}{\left(a_{1}+b_{2}\right)\left(b_{1}-q_{1}\right)}\right],$
$\displaystyle\mathcal{I}(B\,\text{Ir}(B),A_{1})$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{B\,\text{Ir}(B)\cup
B_{1}\,\text{Ir}(B_{1})}+\tilde{S}_{AB\text{Ir}(B)}-\tilde{S}_{A}-\tilde{S}_{B_{1}\text{Ir}(B_{1})}\right)$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{(-\infty,q_{1}]\cup[b_{2},\infty)}+\tilde{S}_{[-a_{4},-a_{1}]\cup[b_{1},b_{4}]}-\tilde{S}_{[b_{1},b_{2}]}-\tilde{S}_{[-a_{1},q_{1}]\cup(-\infty,-a_{4}]\cup[b_{4},\infty)}\right)$
$\displaystyle=\frac{c}{6}\log\left[\frac{(b_{1}+a_{1})(b_{2}-q_{1})}{(b_{2}-b_{1})(a_{1}+q_{1})}\right],$

where $a_{1}=b_{1}+2x_{0}$. Then the total crossing PEE is

$\displaystyle\frac{1}{2}\Big{[}\mathcal{I}(A,B_{1}\text{Ir}(B_{1}))+\mathcal{I}(B\,\text{Ir}(B),A_{1})\Big{]}$
$\displaystyle=\frac{c}{12}\log\left[\frac{(b_{1}+a_{1})^{2}(b_{2}-q_{1})^{2}}{(b_{2}-b_{1})(a_{1}+q_{1})\left(a_{1}+b_{2}\right)\left(b_{1}-q_{1}\right)}\right].$
(124)

The minimal point for the total crossing PEE is

$q_{1}=\frac{b_{1}^{2}}{b_{2}}-\frac{(b_{1}-b_{2})^{2}}{b_{2}^{2}}x_{0},$
(125)

which is exactly the balanced point (87). Furthermore, the minimized crossing
PEE for this adjacent phase is

$\Big{[}\mathcal{I}(A,B_{1}\text{Ir}(B_{1}))+\mathcal{I}(B\,\text{Ir}(B),A_{1})\Big{]}\Big{|}_{\rm
minimal}=\frac{c}{3}\log 2,$ (126)

which is universal and gives the lower bound of the Markov gap. In island
phase, the PEE for adjacent intervals does not recover the mutual information,
and accordingly, the crossing PEE no longer exactly recovers the Markov gap
[2].

For phase D2a, we have four crossing PEEs

$\displaystyle\mathcal{I}(A,B_{1}\mathrm{Ir}(B_{1}))$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{AA_{1}}+\tilde{S}_{AA_{2}B_{2}B\text{Ir}(B)}-\tilde{S}_{A_{1}}-\tilde{S}_{A_{2}B_{2}B\,\text{Ir}(B)}\right)$
(127)
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{[q_{1},b_{2}]}+\tilde{S}_{[-a_{4},-a_{1}]\cup[b_{1},b_{4}]}-\tilde{S}_{[q_{1},b_{1}]}-\tilde{S}_{[-a_{4},-a_{1}]\cup[b_{2},b_{4}]}\right)$
$\displaystyle=\frac{c}{6}\log\frac{(b_{1}+a_{1})(b_{2}-q_{1})}{(a_{1}+b_{2})(b_{1}-q_{1})},$
$\displaystyle\mathcal{I}(A,B_{2})$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{AA_{2}}+\tilde{S}_{B\text{Ir}(B)B_{1}\text{Ir}(B_{1})A_{1}A}-\tilde{S}_{A_{2}}-\tilde{S}_{B\text{Ir}(B)B_{1}\text{Ir}(B_{1})A_{1}}\right)$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{[b_{1},q_{2}]}+\tilde{S}_{[b_{2},b_{3}]}-\tilde{S}_{[b_{2},q_{2}]}-\tilde{S}_{[b_{1},b_{3}]}\right)$
$\displaystyle=\frac{c}{6}\log\frac{(b_{3}-b_{2})(q_{2}-b_{1})}{(b_{3}-b_{1})(q_{2}-b_{2})},$

and

$\displaystyle\mathcal{I}(B\mathrm{Ir}(B),A_{1})$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{AA_{2}B_{2}B\text{Ir}(B)}+\tilde{S}_{B\text{Ir}(B)B_{1}\text{Ir}(B_{1})}-\tilde{S}_{AA_{2}B_{2}}-\tilde{S}_{B_{1}\text{Ir}(B_{1})}\right)$
(128)
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{[-a_{4},-a_{1}]\cup[b_{1},b_{4}]}+\tilde{S}_{[q_{1},b_{3}]}-\tilde{S}_{[b_{1},b_{3}]}-\tilde{S}_{[-\infty,-a_{4}]\cup[-a_{1},q_{1}]\cup[b_{4},\infty]}\right)$
$\displaystyle=\frac{c}{6}\log\frac{(a_{1}+b_{1})(b_{3}-q_{1})}{(a_{1}+q_{1})(b_{3}-b_{1})},$
$\displaystyle\mathcal{I}(B\mathrm{Ir}(B),A_{2})$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{B_{2}B\text{Ir}(B)}+\tilde{S}_{B\text{Ir}(B)B_{1}\text{Ir}(B_{1})A_{1}A}-\tilde{S}_{B_{2}}-\tilde{S}_{B_{1}\text{Ir}(B_{1})A_{1}A}\right)$
$\displaystyle=\frac{1}{2}\left(\tilde{S}_{[-a_{4},-a_{1}]\cup[q_{2},b_{4}]}+\tilde{S}_{[b_{2},b_{3}]}-\tilde{S}_{[q_{2},b_{3}]}-\tilde{S}_{[-a_{4},-a_{1}]\cup[b_{2},b_{4}]}\right)$
$\displaystyle=\frac{c}{6}\log\frac{(q_{2}+a_{1})(b_{3}-b_{2})}{(a_{1}+b_{2})(b_{3}-q_{2})},$

Then the total crossing PEE is

$\displaystyle\mathcal{I}(A,B_{1}\mathrm{Ir}(B_{1}))+\mathcal{I}(A,B_{2})+\mathcal{I}(B\mathrm{Ir}(B),A_{1})+\mathcal{I}(B\mathrm{Ir}(B),A_{2})$
(129) $\displaystyle=$
$\displaystyle\frac{c}{6}\log\left[\frac{(a_{1}+b_{1})^{2}(b_{3}-b_{2})^{2}(q_{2}-b_{1})(q_{2}+a_{1})(b_{2}-q_{1})(b_{3}-q_{1})}{(b_{2}+a_{1})^{2}(b_{3}-b_{1})^{2}(b_{1}-q_{1})(a_{1}+q_{1})(q_{2}-b_{2})(b_{3}-q_{2})}\right].$

The minimal points are given by

$\begin{split}q_{1}&\approx\frac{b_{1}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{1}^{2}+b_{2}b_{3}-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{1}(b_{1}+b_{2})(b_{1}+b_{3})+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}(2b_{1}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{1}+b_{2})(b_{1}+b_{3})}\right)x_{0},\end{split}$
(130)
$\begin{split}q_{2}&\approx\frac{b_{1}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}}{b_{2}+b_{3}}-\left(\frac{2(b_{1}^{2}+b_{2}b_{3}+\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})})}{(b_{2}+b_{3})^{2}}\right.\\\
&\left.+\frac{-2b_{1}(b_{1}+b_{2})(b_{1}+b_{3})-\sqrt{(b_{2}^{2}-b_{1}^{2})(b_{3}^{2}-b_{1}^{2})}(2b_{1}+b_{2}+b_{3})}{(b_{2}+b_{3})(b_{1}+b_{2})(b_{1}+b_{3})}\right)x_{0},\end{split}$
(131)

which is exactly the balanced points (104) and (105).

##  6 Discussion

In this paper we simulate the AdS/BCFT correspondence with a fluctuating KR
brane via a holographic Weyl transformed CFT2, where the Weyl transformation
is adapted to induce a cutoff brane that overlaps with the KR brane. We
calculated the entanglement entropy in the Weyl transformed CFT2 and the
corresponding RT surface and find it coincides with the RT surface in
AdS/BCFT. Furthermore, we calculated the BPE in the Weyl transformed CFT by
taking into account of the island contributions, and find its correspondence
to the EWCS in the holographic picture. Compared with the calculation in [2],
the fluctuation of the brane changes both of the BPE and the EWCS. Provided
different assignments for the ownerless island region, we solve the balance
requirements and find that, the corresponding BPE matches the area of
different saddle points of the EWCS. According to the minimal requirement for
the BPE, we ought to select the assignment that yields the minimal BPE, which
corresponds to the area of the EWCS.

Although the simulation is quite successful in reproducing the RT surfaces
anchored on the brane and the EWCS in the entanglement wedges via field theory
calculations in the Weyl transformed CFT, there are important differences
between the holographic Weyl transformed CFT and the AdS/BCFT set-up.

  * •

Firstly, the Weyl transformed CFT is always settled at the asymptotic
boundary, while the $2d$ effective theory in AdS/BCFT locates on the KR brane
and the bath CFT.

  * •

Secondly, the spacetime behind the cutoff brane is not empty. The cutoff brane
is just the boundary of the region where the RT curves are cut off. We stress
that, when calculating a two-point function of the twist operators, the
corresponding RT curve should always be cut off at some point on the
corresponding cutoff sphere, which in general is not the point on the cutoff
brane. For an arbitrary two-point function, the corresponding RT curve are cut
off at some point behind the cutoff brane, and only the RT curves representing
the optimized two-point functions are cut off at the cutoff brane.

Compared with the AdS/BCFT (or JT gravity coupled to a CFT bath) set-up, the
holographic Weyl transformed CFT2 has many advantages:

  * •

The two-point functions of twist operators without optimization are subtle in
the AdS/BCFT set-up, while they are well defined in the Weyl transformed CFT.
It was pointed out that, the non-optimized two-point functions should not be
understood as an entanglement entropy due to the self-encoding property,
rather it is proposed to be a PEE between the interval bounded by the two
points and its complement [1].

  * •

In the AdS/BCFT set-up, since the JT gravity lives on the brane, the insertion
point of the twist operator in the JT gravity and the point where the RT
surface anchors on the brane should coincide. This is inconsistent with our
observation that $a_{\rm brane}\neq a_{\rm bdy}$, which happens in both of the
AdS/BCFT and the Weyl transformed CFT set-ups. Nevertheless, the inequality is
well understood in the Weyl transformed CFT set-up, as the Weyl transformed
CFT locates at the asymptotic boundary, rather than the brane.

  * •

The Weyl factor 1 (36) is a result of the conjecture that the $2d$ effective
theory in AdS/BCFT lives on the brane. Nevertheless, we showed that the island
formula (2) with the Weyl factor (36) gives us the entanglement entropy (40),
which deviates from the result (49) in doubly holography. While the island
formula with the Weyl factor 2 (41) reproduces (49).

In [21], the authors calculated the entanglement entropy for intervals in the
bath region by the island formula (2) with the Weyl factor 1 (36), and tried
to match it to the result of a bulk formula, which they call the defect
extremal surface (DES) formula. Nevertheless in the set-up of [21], it is
necessary to put specific defect theory on the brane by hand, and the coupling
between the defect theory and the CFT on the bath region to form a $2d$
effective theory is subtle. It will be interesting to decompose the DES set-up
using the Weyl transformed CFT set-up.

The particular Weyl transformations we have used are derived from the
requirement that, the cutoff brane should coincide with the KR brane in the
AdS/BCFT setup. Note that, in the AdS/BCFT set-up the location of the KR brane
is determined by the boundary conditions and tension on the brane and the
Einstein equation of the gravity. It will be very important to explore whether
there is any intrinsic reason in the Weyl transformed CFT that reproduces the
particular Weyl transformation.

In summary, our discussion on the configuration with a fluctuating brane
indicates that, the Weyl transformed CFT gives us a more consistent
interpretation for the island formulas of entanglement entropy and BPE, which
further indicates that several features of the original AdS/BCFT setup should
be reconsidered. Furthermore, the differences between the Weyl transformed CFT
and the AdS/BCFT setup indicates that the $2d$ effective field theory should
locate on the asymptotic boundary rather than the KR brane, which resolves the
puzzle of $a_{\rm bdy}\neq a_{\rm brane}$ in the AdS/BCFT setup. Exploring the
differences between these two setups and testing their self-consistency from
other perspectives will be very interesting future directions.

## Acknowledgment

We would like to thank Ziming Ji, Zhenbin Yang and Yang Zhou for helpful
discussions. J. Lin is supported by the National Natural Science Foundation of
China under Grant No.12247117, No.12247103 and No.12047502. Y. Lu receives
support from the China Postdoctoral Science Foundation under Grant
No.2022TQ0140, the National Natural Science Foundation of China under Grant
No.12247161, and the NSFC Research Fund for International Scientists (Grant
No. 12250410250).

##  Appendix A Cutoff brane of Weyl transformed CFT

Figure 15: Cutoff brane (the red curve) in Weyl transformed CFT is the comment
tangent line of the set of cutoff spheres.

We first solve the cutoff sphere associated with $x=-x_{1}$, which is defined
as a set of points with the geodesic distance $|\varphi(-x_{1})|$ from the
boundary $x=-x_{1}$, where

$\varphi(x)=\begin{cases}0&\text{ if }\quad x>0\\\
-\log\left(\frac{2|x|}{\epsilon}\right)+\kappa_{1}+\frac{\kappa_{2}}{|x|}&\
\text{if }\quad x<0\end{cases}$ (132)

The Poincaré AdS3 metric is

$\mathrm{d}s^{2}=\frac{-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}z^{2}}{z^{2}}.$
(133)

Using the geodesic length between two spacelike-separated points [80]

$\displaystyle L_{\text{AdS
}}\left(U_{1},V_{1},\rho_{1},U_{2},V_{2},\rho_{2}\right)$ (134)
$\displaystyle\quad=\frac{1}{2}\log\left[\frac{\rho_{2}\left(\rho_{2}+X\right)+\rho_{1}\left(\rho_{2}Y\left(2\rho_{2}+X\right)+X\right)+\left(\rho_{1}+\rho_{2}\rho_{1}Y\right)^{2}}{2\rho_{1}\rho_{2}}\right],$

where

$U=\frac{x+t}{2},\quad V=\frac{x-t}{2},\quad\rho=\frac{2}{z^{2}},$ (135)

are light-cone coordinate and

$\displaystyle Y=2\left(U_{1}-U_{2}\right)\left(V_{1}-V_{2}\right),\quad
X=\sqrt{\rho_{1}^{2}+2\rho_{2}\rho_{1}\left(\rho_{1}Y-1\right)+\left(\rho_{2}+\rho_{1}\rho_{2}Y\right)^{2}},$
(136)

we get the cutoff sphere associated with $x=-x_{1}$ on a static time slice
$t=0$

$\left(x-x_{1}\right)^{2}+\left(z-x_{1}e^{-\kappa_{1}-\frac{\kappa_{2}}{|x_{1}|}}\right)^{2}=x_{1}^{2}e^{-2\kappa_{1}-2\frac{\kappa_{2}}{x_{1}}},$
(137)

which is just the circle at
$(-x_{1},x_{1}e^{-\kappa_{1}-\frac{\kappa_{2}}{x_{1}}})$ with radius
$r=e^{-\kappa_{1}-\frac{\kappa_{2}}{x_{1}}}$.

Now we seek for the common tangent line $C^{1}$ of the set of cutoff spheres
(see the illustration in Fig.15). This tangent line $C^{1}$ is what we call
cutoff surface for Weyl transformed CFT. The curve $C^{0}$, which is made up
of the set of the center points of all the cutoff spheres, is given by

$C^{0}:\quad z=e^{-\kappa_{1}-\frac{\kappa_{2}}{|x|}}.$ (138)

Then the tangent line of $C^{0}$ at $O=(-x_{1},r)$ and the tangent line of
$C^{1}$ at $Q=(-x_{m},z_{m})$ will both intersect with the AdS boundary at

$x=-x_{2}=-\frac{\kappa_{2}x_{1}}{\kappa_{2}+x_{1}},$ (139)

with the intersection angle $\alpha$ and $2\alpha$, respectively and

$\tan\alpha=e^{-(\kappa_{1}+\frac{\kappa_{2}}{x_{1}})}\left(\frac{\kappa_{2}}{x_{1}}+1\right).$
(140)

Then the tangent point $Q=(-x_{m},z_{m})$ is determined by the following
geometric relations

$\frac{z_{m}}{x_{m}-x_{2}}=\tan
2\alpha,\quad\frac{x_{m}-x_{2}}{x_{1}-x_{2}}=\cos 2\alpha.$ (141)

Solving (141), we arrive at the parametric equation of the cutoff surface

$\begin{split}-x_{m}&\approx-
x_{1}\tanh{\kappa_{1}}-(1-\tanh{\kappa_{1}})\kappa_{2}+\mathcal{O}(\kappa_{2}^{2}),\\\
z_{m}&\approx
x_{1}\cosh^{-1}{\kappa_{1}}-\kappa_{2}\cosh^{-1}{\kappa_{1}}+\mathcal{O}(\kappa_{2}^{2}),\end{split}$
(142)

which is parameterized by $x_{1}$.

Now we show that the location of the cutoff brane (142) coincide with the JT
brane. The location of JT brane is given by

$\begin{split}-x_{J}&=-|y|\tanh\left[\rho_{0}\left(1+\frac{\bar{\phi}_{r}}{|y|}\right)\right]\approx-|y|\tanh\rho_{0}-\frac{\rho_{0}\bar{\phi}_{r}}{\ell}\cosh^{-2}\left(\rho_{0}\right)+\mathcal{O}(\bar{\phi}_{r}^{2})\\\
z_{J}&=|y|\cosh^{-1}\left[\rho_{0}\left(1+\frac{\bar{\phi}_{r}}{|y|}\right)\right]\approx|y|\cosh^{-1}\rho_{0}-\frac{\rho_{0}\bar{\phi}_{r}}{\ell}\cosh^{-1}\left(\rho_{0}\right)\tanh\left(\rho_{0}\right)+\mathcal{O}(\bar{\phi}_{r}^{2}),\end{split}$
(143)

which is parameterized by $|y|=\sqrt{x_{J}^{2}+z_{J}^{2}}$, which is the
Euclidean distance between a point $(-x_{J},z_{J})$ on the brane and the
origin point. To compare JT brane (143) with the cutoff brane (142), let us
reparametrize the cutoff brane in terms of $|y|$. Using

$\begin{split}|y|=\sqrt{x_{m}^{2}+z_{m}^{2}}\approx
x_{1}+\kappa_{2}(\tanh\kappa_{1}-1)+\mathcal{O}(\kappa_{2}^{2}),\end{split}$
(144)

we get the reparametrized cutoff brane

$\begin{split}-x_{m}&\approx-|y|\tanh{\kappa_{1}}-\kappa_{2}\cosh^{-2}\kappa_{1}+\mathcal{O}(\kappa_{2}^{2}),\\\
z_{m}&\approx|y|\cosh^{-1}{\kappa_{1}}-\kappa_{2}\tanh\kappa_{1}\cosh^{-1}\kappa_{1}+\mathcal{O}(\kappa_{2}^{2}).\end{split}$
(145)

Identifying

$\kappa_{1}=\rho_{0},\quad\kappa_{2}=\frac{\rho_{0}\bar{\phi}_{r}}{\ell},$
(146)

then the cutoff brane coincides with the JT brane.

##  Appendix B A trick to calculate EWCS for phase D2b

Figure 16: The area of EWCS can be obtained by the area of $\Gamma$ minus the
area of $\Gamma_{1}$, where $\Gamma$ is the extended EWCS and $\Gamma_{1}$ the
extended part of $\Gamma$.

Let us extend EWCS for phase D2b to the point $(\tilde{q}_{2},0)$ on the AdS
boundary (see Fig.16). Then this extended EWCS $\Gamma$ is the RT surface
connecting AdS boundary point $x=\tilde{q}_{2}$ and the point $y=\tilde{q}$ on
the brane and its area is

$\begin{split}\frac{\text{Area}(\Gamma)}{4G_{N}}=&\frac{c}{6}\log\frac{\tilde{q}_{2}^{2}+2\tilde{q}_{2}\tilde{q}\tanh\rho_{0}+\tilde{q}^{2}}{\tilde{q}_{2}\epsilon\cosh^{-1}\rho_{0}}+\frac{cx_{0}}{6\tilde{q}},\end{split}$
(147)

where

$\tilde{q}\approx\tilde{q}_{2}+x_{0}\left(1+\tanh\rho_{0}\right).$ (148)

Note that the EWCS is minimal curve and thus perpendicular to the RT surface
associated with the interval $[b_{2},b_{3}]$. Then, the area of the segment
$\Gamma_{1}$ on $\Gamma$ that starts from $x=\tilde{q}_{2}$ and ends at the RT
surface for $[b_{2},b_{3}]$ is given by

$\begin{split}\frac{\text{Area}(\Gamma_{1})}{4G_{N}}=&\frac{c}{6}\log\frac{2(\tilde{q}_{2}-b_{2})(b_{3}-\tilde{q}_{2})}{(b_{3}-b_{2})\epsilon}.\end{split}$
(149)

Finally, the area of EWCS is calculated by the following extremization
procedure

$\frac{\text{Area}(\text{EWCS})}{4G_{N}}=\text{Ext}_{\tilde{q}_{2}}\left(\frac{\text{Area}(\Gamma)}{4G_{N}}-\frac{\text{Area}(\Gamma_{1})}{4G_{N}}\right),$
(150)

with the extremal point is

$\tilde{q}_{2}=\sqrt{b_{2}b_{3}}+3(\sqrt{b_{3}}-\sqrt{b_{2}})^{2}\frac{x_{0}}{6\sqrt{b_{2}b_{3}}}.$
(151)

Then the area of EWCS is

$\begin{split}\frac{\text{Area}(\text{EWCS})}{4G_{N}}=\frac{c}{6}\log\frac{\sqrt{b_{3}}+\sqrt{b_{2}}}{\sqrt{b_{3}}-\sqrt{b_{2}}}+\frac{c}{6}\rho_{0}+\frac{c}{6}\frac{x_{0}}{\sqrt{b_{2}b_{3}}}.\end{split}$
(152)

## References

  * [1] D. Basu, Q. Wen and S. Zhou,  _Entanglement Islands from Hilbert Space Reduction_ (2022),  2211.17004.
  * [2] D. Basu, J. Lin, Y. Lu and Q. Wen,  _Ownerless island and partial entanglement entropy in island phases_ (2023),  2305.04259.
  * [3] J. M. Maldacena,  _The Large N limit of superconformal field theories and supergravity_ ,  Adv. Theor. Math. Phys. 2, 231 (1998),  10.4310/ATMP.1998.v2.n2.a1,  hep-th/9711200.
  * [4] S. Ryu and T. Takayanagi,  _Holographic derivation of entanglement entropy from AdS/CFT_ ,  Phys. Rev. Lett. 96, 181602 (2006),  10.1103/PhysRevLett.96.181602,  hep-th/0603001.
  * [5] V. E. Hubeny, M. Rangamani and T. Takayanagi,  _A Covariant holographic entanglement entropy proposal_ ,  JHEP 07, 062 (2007),  10.1088/1126-6708/2007/07/062,  0705.0016.
  * [6] A. Lewkowycz and J. Maldacena,  _Generalized gravitational entropy_ ,  JHEP 08, 090 (2013),  10.1007/JHEP08(2013)090,  1304.4926.
  * [7] N. Engelhardt and A. C. Wall,  _Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime_ ,  JHEP 01, 073 (2015),  10.1007/JHEP01(2015)073,  1408.3203.
  * [8] A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield,  _The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole_ ,  JHEP 12, 063 (2019),  10.1007/JHEP12(2019)063,  1905.08762.
  * [9] G. Penington,  _Entanglement Wedge Reconstruction and the Information Paradox_ ,  JHEP 09, 002 (2020),  10.1007/JHEP09(2020)002,  1905.08255.
  * [10] A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao,  _The Page curve of Hawking radiation from semiclassical geometry_ ,  JHEP 03, 149 (2020),  10.1007/JHEP03(2020)149,  1908.10996.
  * [11] G. Penington, S. H. Shenker, D. Stanford and Z. Yang,  _Replica wormholes and the black hole interior_ ,  JHEP 03, 205 (2022),  10.1007/JHEP03(2022)205,  1911.11977.
  * [12] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini,  _Replica Wormholes and the Entropy of Hawking Radiation_ ,  JHEP 05, 013 (2020),  10.1007/JHEP05(2020)013,  1911.12333.
  * [13] D. Marolf and H. Maxfield,  _Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information_ ,  JHEP 08, 044 (2020),  10.1007/JHEP08(2020)044,  2002.08950.
  * [14] Y. Lu and J. Lin,  _The Markov gap in the presence of islands_ ,  JHEP 03, 043 (2023),  10.1007/JHEP03(2023)043,  2211.06886.
  * [15] Y. Lu and J. Lin,  _Islands in Kaluza–Klein black holes_ ,  Eur. Phys. J. C 82(2), 132 (2022),  10.1140/epjc/s10052-022-10074-w,  2106.07845.
  * [16] J. Kumar Basak, D. Basu, V. Malvimat, H. Parihar and G. Sengupta,  _Islands for entanglement negativity_ ,  SciPost Phys. 12(1), 003 (2022),  10.21468/SciPostPhys.12.1.003,  2012.03983.
  * [17] M.-H. Yu, X.-H. Ge and C.-Y. Lu,  _Page Curves for Accelerating Black Holes_ (2023),  2306.11407.
  * [18] R.-X. Miao,  _Massless Entanglement Island in Wedge Holography_ (2022),  2212.07645.
  * [19] D. Li and R.-X. Miao,  _Massless entanglement islands in cone holography_ ,  JHEP 06, 056 (2023),  10.1007/JHEP06(2023)056,  2303.10958.
  * [20] H. Geng and A. Karch,  _Massive islands_ ,  JHEP 09, 121 (2020),  10.1007/JHEP09(2020)121,  2006.02438.
  * [21] F. Deng, J. Chu and Y. Zhou,  _Defect extremal surface as the holographic counterpart of Island formula_ ,  JHEP 03, 008 (2021),  10.1007/JHEP03(2021)008,  2012.07612.
  * [22] Y. An and P. Cheng,  _Replica wormhole as a vacuum-to-vacuum transition_ ,  Eur. Phys. J. C 83(4), 341 (2023),  10.1140/epjc/s10052-023-11518-7,  2304.09432.
  * [23] W.-C. Gan, D.-H. Du and F.-W. Shu,  _Island and Page curve for one-sided asymptotically flat black hole_ ,  JHEP 07, 020 (2022),  10.1007/JHEP07(2022)020,  2203.06310.
  * [24] T. Hartman, Y. Jiang and E. Shaghoulian,  _Islands in cosmology_ ,  JHEP 11, 111 (2020),  10.1007/JHEP11(2020)111,  2008.01022.
  * [25] K. Hashimoto, N. Iizuka and Y. Matsuo,  _Islands in Schwarzschild black holes_ ,  JHEP 06, 085 (2020),  10.1007/JHEP06(2020)085,  2004.05863.
  * [26] Y. Ling, Y. Liu and Z.-Y. Xian,  _Island in Charged Black Holes_ ,  JHEP 03, 251 (2021),  10.1007/JHEP03(2021)251,  2010.00037.
  * [27] I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi and Z. Wei,  _Entanglement Entropy in a Holographic Moving Mirror and the Page Curve_ ,  Phys. Rev. Lett. 126(6), 061604 (2021),  10.1103/PhysRevLett.126.061604,  2011.12005.
  * [28] X. Wang, R. Li and J. Wang,  _Islands and Page curves of Reissner-Nordström black holes_ ,  JHEP 04, 103 (2021),  10.1007/JHEP04(2021)103,  2101.06867.
  * [29] Y. Guo and R.-X. Miao,  _Page curves on codim-m and charged branes_ ,  Eur. Phys. J. C 83(9), 847 (2023),  10.1140/epjc/s10052-023-12026-4.
  * [30] J.-C. Chang, S. He, Y.-X. Liu and L. Zhao,  _Island formula in Planck brane_ ,  JHEP 11, 006 (2023),  10.1007/JHEP11(2023)006,  2308.03645.
  * [31] M. Afrasiar, J. K. Basak, A. Chandra and G. Sengupta,  _Reflected entropy for communicating black holes. Part I. Karch-Randall braneworlds_ ,  JHEP 02, 203 (2023),  10.1007/JHEP02(2023)203,  2211.13246.
  * [32] H. Geng, A. Karch, C. Perez-Pardavila, L. Randall, M. Riojas, S. Shashi and M. Youssef,  _Constraining braneworlds with entanglement entropy_ (2023),  2306.15672.
  * [33] F. Deng, Z. Wang and Y. Zhou,  _End of the World Brane meets $T\bar{T}$_ (2023),  2310.15031.
  * [34] K. Suzuki and T. Takayanagi,  _BCFT and Islands in two dimensions_ ,  JHEP 06, 095 (2022),  10.1007/JHEP06(2022)095,  2202.08462.
  * [35] L. Randall and R. Sundrum,  _A Large mass hierarchy from a small extra dimension_ ,  Phys. Rev. Lett. 83, 3370 (1999),  10.1103/PhysRevLett.83.3370,  hep-ph/9905221.
  * [36] L. Randall and R. Sundrum,  _An Alternative to compactification_ ,  Phys. Rev. Lett. 83, 4690 (1999),  10.1103/PhysRevLett.83.4690,  hep-th/9906064.
  * [37] A. Karch and L. Randall,  _Locally localized gravity_ ,  JHEP 05, 008 (2001),  10.1088/1126-6708/2001/05/008,  hep-th/0011156.
  * [38] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi,  _Jackiw-Teitelboim Gravity from the Karch-Randall Braneworld_ ,  Phys. Rev. Lett. 129(23), 231601 (2022),  10.1103/PhysRevLett.129.231601,  2206.04695.
  * [39] H. Geng,  _Aspects of AdS 2 quantum gravity and the Karch-Randall braneworld_,  JHEP 09, 024 (2022),  10.1007/JHEP09(2022)024,  2206.11277.
  * [40] F. Deng, Y.-S. An and Y. Zhou,  _JT gravity from partial reduction and defect extremal surface_ ,  JHEP 02, 219 (2023),  10.1007/JHEP02(2023)219,  2206.09609.
  * [41] T. Faulkner, A. Lewkowycz and J. Maldacena,  _Quantum corrections to holographic entanglement entropy_ ,  JHEP 11, 074 (2013),  10.1007/JHEP11(2013)074,  1307.2892.
  * [42] T. Takayanagi,  _Holographic Dual of BCFT_ ,  Phys. Rev. Lett. 107, 101602 (2011),  10.1103/PhysRevLett.107.101602,  1105.5165.
  * [43] B. M. Terhal, M. Horodecki, D. W. Leung and D. P. DiVincenzo,  _The entanglement of purification_ ,  Journal of Mathematical Physics 43, 4286–4298 (2002),  https://doi.org/10.1063/1.1498001.
  * [44] T. Takayanagi and K. Umemoto,  _Entanglement of purification through holographic duality_ ,  Nature Phys. 14(6), 573 (2018),  10.1038/s41567-018-0075-2,  1708.09393.
  * [45] P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel and B. Swingle,  _Entanglement of purification: from spin chains to holography_ ,  JHEP 01, 098 (2018),  10.1007/JHEP01(2018)098,  1709.07424.
  * [46] J. Kudler-Flam and S. Ryu,  _Entanglement negativity and minimal entanglement wedge cross sections in holographic theories_ ,  Phys. Rev. D 99(10), 106014 (2019),  10.1103/PhysRevD.99.106014,  1808.00446.
  * [47] Y. Kusuki, J. Kudler-Flam and S. Ryu,  _Derivation of holographic negativity in AdS 3/CFT2_,  Phys. Rev. Lett. 123(13), 131603 (2019),  10.1103/PhysRevLett.123.131603,  1907.07824.
  * [48] P. Chaturvedi, V. Malvimat and G. Sengupta,  _Holographic Quantum Entanglement Negativity_ ,  JHEP 05, 172 (2018),  10.1007/JHEP05(2018)172,  1609.06609.
  * [49] S. Dutta and T. Faulkner,  _A canonical purification for the entanglement wedge cross-section_ ,  JHEP 03, 178 (2021),  10.1007/JHEP03(2021)178,  1905.00577.
  * [50] K. Tamaoka,  _Entanglement Wedge Cross Section from the Dual Density Matrix_ ,  Phys. Rev. Lett. 122(14), 141601 (2019),  10.1103/PhysRevLett.122.141601,  1809.09109.
  * [51] R. Espíndola, A. Guijosa and J. F. Pedraza,  _Entanglement Wedge Reconstruction and Entanglement of Purification_ ,  Eur. Phys. J. C 78(8), 646 (2018),  10.1140/epjc/s10052-018-6140-2,  1804.05855.
  * [52] C. A. Agón, J. De Boer and J. F. Pedraza,  _Geometric Aspects of Holographic Bit Threads_ ,  JHEP 05, 075 (2019),  10.1007/JHEP05(2019)075,  1811.08879.
  * [53] J. Levin, O. DeWolfe and G. Smith,  _Correlation measures and distillable entanglement in AdS/CFT_ ,  Phys. Rev. D 101(4), 046015 (2020),  10.1103/PhysRevD.101.046015,  1909.04727.
  * [54] Q. Wen,  _Balanced Partial Entanglement and the Entanglement Wedge Cross Section_ ,  JHEP 04, 301 (2021),  10.1007/JHEP04(2021)301,  2103.00415.
  * [55] Q. Wen and H. Zhong,  _Covariant entanglement wedge cross-section, balanced partial entanglement and gravitational anomalies_ ,  SciPost Phys. 13(3), 056 (2022),  10.21468/SciPostPhys.13.3.056,  2205.10858.
  * [56] H. A. Camargo, P. Nandy, Q. Wen and H. Zhong,  _Balanced partial entanglement and mixed state correlations_ ,  SciPost Phys. 12(4), 137 (2022),  10.21468/SciPostPhys.12.4.137,  2201.13362.
  * [57] D. Basu,  _Balanced Partial Entanglement in Flat Holography_ (2022),  2203.05491.
  * [58] A. Bagchi,  _Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories_ ,  Phys. Rev. Lett. 105, 171601 (2010),  10.1103/PhysRevLett.105.171601,  1006.3354.
  * [59] H. Jiang, W. Song and Q. Wen,  _Entanglement Entropy in Flat Holography_ ,  JHEP 07, 142 (2017),  10.1007/JHEP07(2017)142,  1706.07552.
  * [60] A. Bhattacharya, A. Bhattacharyya and A. K. Patra,  _Holographic complexity of Jackiw-Teitelboim gravity from Karch-Randall braneworld_ ,  JHEP 07, 060 (2023),  10.1007/JHEP07(2023)060,  2304.09909.
  * [61] S. E. Aguilar-Gutierrez, A. K. Patra and J. F. Pedraza,  _Entangled universes in dS wedge holography_ ,  JHEP 10, 156 (2023),  10.1007/JHEP10(2023)156,  2308.05666.
  * [62] Y. Chen and G. Vidal,  _Entanglement contour_ ,  Journal of Statistical Mechanics: Theory and Experiment p. P10011 (2014),  https://doi.org/10.48550/arXiv.1406.1471,  1406.1471.
  * [63] Q. Wen,  _Fine structure in holographic entanglement and entanglement contour_ ,  Phys. Rev. D 98(10), 106004 (2018),  10.1103/PhysRevD.98.106004,  1803.05552.
  * [64] J. Kudler-Flam, I. MacCormack and S. Ryu,  _Holographic entanglement contour, bit threads, and the entanglement tsunami_ ,  J. Phys. A 52(32), 325401 (2019),  10.1088/1751-8121/ab2dae,  1902.04654.
  * [65] M. Han and Q. Wen,  _Entanglement entropy from entanglement contour: higher dimensions_ ,  SciPost Phys. Core 5, 020 (2022),  10.21468/SciPostPhysCore.5.2.020,  1905.05522.
  * [66] Q. Wen,  _Formulas for Partial Entanglement Entropy_ ,  Phys. Rev. Res. 2(2), 023170 (2020),  10.1103/PhysRevResearch.2.023170,  1910.10978.
  * [67] Q. Wen,  _Entanglement contour and modular flow from subset entanglement entropies_ ,  JHEP 05, 018 (2020),  10.1007/JHEP05(2020)018,  1902.06905.
  * [68] M. Han and Q. Wen,  _First law and quantum correction for holographic entanglement contour_ ,  SciPost Phys. 11(3), 058 (2021),  10.21468/SciPostPhys.11.3.058,  2106.12397.
  * [69] D. S. Ageev,  _Shaping contours of entanglement islands in BCFT_ ,  JHEP 03, 033 (2022),  10.1007/JHEP03(2022)033,  2107.09083.
  * [70] A. Rolph,  _Local measures of entanglement in black holes and CFTs_ ,  SciPost Phys. 12(3), 079 (2022),  10.21468/SciPostPhys.12.3.079,  2107.11385.
  * [71] Y.-Y. Lin, J.-R. Sun, Y. Sun and J.-C. Jin,  _The PEE aspects of entanglement islands from bit threads_ ,  JHEP 07, 009 (2022),  10.1007/JHEP07(2022)009,  2203.03111.
  * [72] Y.-Y. Lin,  _Distilled density matrices of holographic partial entanglement entropy from thread-state correspondence_ ,  Phys. Rev. D 108(10), 106010 (2023),  10.1103/PhysRevD.108.106010,  2305.02895.
  * [73] Y. Liu,  _Revisiting the 3D flat Holography: Causality structure and modular flow_ (2023),  2309.05220.
  * [74] J. Lin, Y. Lu and Q. Wen,  _Geometrizing the Partial Entanglement Entropy: from PEE Threads to Bit Threads_ (2023),  2311.02301.
  * [75] H. Casini and M. Huerta,  _Remarks on the entanglement entropy for disconnected regions_ ,  JHEP 03, 048 (2009),  10.1088/1126-6708/2009/03/048,  0812.1773.
  * [76] Y. Zou, K. Siva, T. Soejima, R. S. K. Mong and M. P. Zaletel,  _Universal tripartite entanglement in one-dimensional many-body systems_ ,  Phys. Rev. Lett. 126(12), 120501 (2021),  10.1103/PhysRevLett.126.120501,  2011.11864.
  * [77] P. Hayden, O. Parrikar and J. Sorce,  _The Markov gap for geometric reflected entropy_ ,  JHEP 10, 047 (2021),  10.1007/JHEP10(2021)047,  2107.00009.
  * [78] J. Maldacena, D. Stanford and Z. Yang,  _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ ,  PTEP 2016(12), 12C104 (2016),  10.1093/ptep/ptw124,  1606.01857.
  * [79] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi,  _Information Transfer with a Gravitating Bath_ ,  SciPost Phys. 10(5), 103 (2021),  10.21468/SciPostPhys.10.5.103,  2012.04671.
  * [80] Q. Wen,  _Towards the generalized gravitational entropy for spacetimes with non-Lorentz invariant duals_ ,  JHEP 01, 220 (2019),  10.1007/JHEP01(2019)220,  1810.11756.