aainstitutetext: Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japanbbinstitutetext:
Inamori Research Institute for Science,
620 Suiginya-cho, Shimogyo-ku,Kyoto 600-8411 Japanccinstitutetext: Yau
Mathematical Sciences Center, Tsinghua University,
Beijing 100084, China

# Non-extremal Island in de Sitter Gravity

Peng-Xiang Hao a Taishi Kawamoto a Shan-Ming Ruan a,b Tadashi Takayanagi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper investigates the challenges and resolutions in computing the
entanglement entropy for the quantum field theory coupled to de Sitter (dS)
gravity along a timelike boundary. The conventional island formula, originally
designed to calculate the fine-grained entropy for a non-gravitational system
coupled to anti-de Sitter (AdS) gravity, encounters difficulties in de Sitter
gravitational spacetime, failing to provide a physically plausible extremal
island. To overcome these problems, we introduce a doubly holographic model by
embedding a dS2 braneworld in an AdS3 bulk spacetime. This approach
facilitates the computation of entanglement entropy through holographic
correlation functions, effectively circumventing the constraints of the island
formula. We demonstrate that the correct recipe for calculating entanglement
entropy with dS gravity involves the non-extremal island, whose boundary is
instead defined at the edge of the dS gravitational region. Our findings
indicate that, during the island phase, the entanglement wedge of the non-
gravitational bath includes the entire dS gravitational space. Using the
second variation formula, we further show that the existence of a locally
minimal surface anchored on the gravitational brane is intrinsically linked to
the extrinsic curvature of the brane.

YITP-24-91

##  1 Introduction

The island formula computes the entanglement entropy in the quantum field
theory coupled to a gravitational theory across its boundary Penington:2019npb
; Almheiri:2019psf ; Almheiri:2019hni . This formula generalizes the
holographic entanglement entropy formula Ryu:2006bv ; Ryu:2006ef ;
Hubeny:2007xt ; Faulkner:2013ana ; Engelhardt:2014gca  with quantum
corrections in the context of the AdS/CFT correspondence Maldacena:1997re . In
the island prescription, the entanglement entropy for a subsystem
$\mathcal{A}$ in the CFT region is calculated by positing an island region
$\mathcal{I}$ in the gravitational theory and extremizing the sum of the
gravitational entropy of the island and the entanglement entropy in the
effective field theory for the subsystem defined by the union of $\mathcal{A}$
and the island $\mathcal{I}$. Notably, the island formula has been
instrumental in deriving the Page curve Page:1993wv , a crucial indicator of
unitarity during black hole evaporation. This was achieved by considering a
conformal field theory (CFT) on a half-space interacting with a gravitational
theory containing a black hole along its timelike boundary Penington:2019npb ;
Almheiri:2019psf ; Almheiri:2019hni . In setups involving two-dimensional
black holes, the island formula is derived directly from the gravitational
path integrals of Jackiw-Teitelboim (JT) gravity, taking into account the non-
perturbative contributions from replica wormholes Almheiri:2019qdq ;
Penington:2019kki .

It is also useful to note that the island formula can be understood by
combining the gravity dual of boundary conformal field theories (BCFTs)
Takayanagi:2011zk ; Fujita:2011fp ; Karch:2000gx  with braneworld holography
Randall:1999ee ; Randall:1999vf ; Gubser:1999vj ; Karch:2000ct , a concept
known as double holography Almheiri:2019hni ; Suzuki:2022xwv . In this doubly
holographic approach, a $d$-dimensional CFT coupled to $d$-dimensional quantum
gravity on an asymptotically AdS space is holographically described by
classical gravity on a $d+1$-dimensional AdS bulk spacetime with an end-of-
the-world brane (EOW brane), which is equivalent to the AdSd+1/BCFTd model. In
the classical limit of the bulk gravity, the entanglement entropy can be
computed as the area of the extremal surface according to the general
holographic formula Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt . In the presence
of EOW branes, a notable feature is that the extremal surface can end on the
EOW brane Takayanagi:2011zk ; Fujita:2011fp , necessitating extremalization
over such configurations. This new twist nicely explains the emergence of
island region on the EOW brane. This idea is illustrated in figure 1. An
important feature in such setups is the presence of a non-trivial extremal
surface $\Gamma$ that departs from the AdS boundary and ends on the EOW brane.
Moreover, the extremal surface $\Gamma$ is stable, as its area is minimized
under spatial deformations and maximized under Lorentzian timelike direction,
which is required for the consistent calculation of holographic entanglement
entropy Hubeny:2007xt ; Wall:2012uf .

Figure 1: The two distinct setups for calculating entanglement entropy with
islands are illustrated here. In the left panel, a CFT is coupled to a
gravitational theory in a AdS space, whereas in the right panel, it is coupled
to a gravitational theory in dS space. The subsystem $\mathcal{A}$, for which
we compute the entanglement entropy, is represented by the yellow horizontal
line. The doubly holographic description, employing braneworld holography, is
depicted on the right side of each panel. The extremal surfaces $\Gamma$,
indicated by the purple curves, each extend from a point $P$ on the AdS
boundary and terminate on the EOW branes. These surfaces describe the
holographic entanglement entropy in the AdS/BCFT correspondence. While the
extremal surface $\Gamma$ can intersect within the AdS brane, it should only
terminate at the edge of the dS brane.

A pertinent question is whether the island formula retains its validity in
more general spacetimes. Extending the application of the island formula to
cosmological spacetimes, such as those with a positive cosmological constant,
could provide important insights into the nature of holography in contexts
more closely aligned with realistic settings beyond the AdS/CFT
correspondence. Additionally, this investigation elucidates the structure of
the Hilbert space in quantum gravity within cosmological spacetimes. In this
paper, we are motivated by this question and investigate a setup wherein a CFT
on one half of de Sitter space is coupled to gravity on the other half across
its timelike boundary. We predominantly utilize the doubly holographic model,
as illustrated in the right panel of figure 1. To ensure analytically
tractable examples, our focus is on a two-dimensional de Sitter space. Upon
straightforward application of the island formula, a perplexing issue arises:
there generally does not exist any physically sensible island that extremizes
the generalized entropy functional in dS gravity. Although we frequently
observe apparent “fake” islands that seem to extremize the total entropy
functional, these solutions are inadmissible for various reasons.

Nevertheless, the correct prescription can be determined through the framework
of double holography. By analyzing holographic computations of one-point
functions in the AdS3/BCFT2 correspondence Suzuki:2022xwv ; Izumi:2022opi , we
can facilitate the computation of entanglement entropy while circumventing the
constraints imposed by the island formula. Our results indicate that the
boundary of non-extremal island is at the edge of the dS gravitational region.
Depending on the size of the subsystem $\mathcal{A}$ in the CFT region, the
island can either be an empty region or encompass the entire (gravitating)
half of de Sitter space. As a result, the entanglement wedge exhibits a
fascinating phase transition phenomenon.

For earlier analyses of the island formula applied to de Sitter spaces, refer
to Chen:2020tes ; Hartman:2020khs ; Balasubramanian:2020xqf ; Sybesma:2020fxg
; Aguilar-Gutierrez:2021bns ; Bousso:2021sji ; Bousso:2022gth ; Geng:2021wcq ;
Kames-King:2021etp ; Aalsma:2021bit ; Teresi:2021qff ; Levine:2022wos ;
Aalsma:2022swk ; Chang:2023gkt ; Balasubramanian:2023xyd ; Franken:2023pni ,
where special features unique to closed universes have been emphasized. A
significant difference between the present work and earlier studies is that
our dS spacetime is not closed, as it has a timelike boundary and can be
directly compared with the familiar setup of islands in AdS black hole
spacetimes. Another advantage of our model is that it allows us to explore a
possible holographic dual of de Sitter space Strominger:2001pn ;
Maldacena:2002vr  (refer to Anninos:2011ui ; Hikida:2021ese ; Susskind:2022bia
; Narovlansky:2023lfz  for explicit examples), which remains poorly
understood. By embedding de Sitter space into a higher-dimensional AdS/CFT via
double holography, we gain more precise insights into the holographic dual of
a half de Sitter space Kawamoto:2023nki , argued to be a highly non-local
field theory.

This paper is organized as follows. In section 2, we apply the island formula
to a scenario in which a CFT2 on one half of dS2 spacetime is coupled to dS
gravity on the other half, highlighting significant issues encountered when
attempting to extremize the entropy functional. Section 2 addresses the
accurate computation of entanglement entropy in the setup described in section
2, utilizing the double holography model. We derive the correct prescription
by examining both the holographic calculation of one-point functions and the
geometric calculation of holographic entanglement entropy. In section 4, we
further demonstrate that the existence of a locally minimal surface anchored
on the EOW brane in AdS space is related to its extrinsic curvature, using the
second variation formula for geodesic variations. Finally, section 5
summarizes our conclusions and outlines directions for future research.

##  2 Invalid island formula for dS gravity

Figure 2: Left: Penrose diagram of an eternal AdS2 black hole coupled to a
non-gravitational thermal bath. Right: Penrose diagram of a global dS2, where
we identify the two north poles (indicated by the left/right black line) to
glue the two copies together. One half of dS space, indicated by the red
color, is gravitating.

As shown in figure 2, our study examines a system incorporating both
gravitational and non-gravitational region. For illustrative purposes, we
consider a thermal bath coupled to an eternal AdS black hole, serving as a
simplified model to demonstrate a Hawking-like information paradox. The island
formula effectively resolves this paradox by utilizing a non-trivial quantum
extremal surface, as detailed in Almheiri:2019yqk . The primary focus of this
paper is the entanglement entropy of a subregion within a non-gravitational
system that is, in contrast, coupled to de Sitter gravity. We anticipate that
the island formula will be essential for calculating the entanglement entropy
in this context. However, our findings reveal that the island formula does not
yield the dominant result, as the unique quantum extremal surface in the de
Sitter gravity configuration does not represent the desired maximin surface
Hubeny:2007xt ; Wall:2012uf .

###  2.1 Half dS2 gravity coupled with CFT2 bath

Figure 3: Left: The Hartle-Hawking state, which includes half of dS2 space as
the gravitational region and the other half as the bath region. The Euclidean
part is represented by the blue hemisphere, while the Lorentzian part is
indicated by the upper hyperboloid. Right: A time slice of the setup for
gluing a half dS2 gravity with a CFT2 living on dS2.

In this paper, we will focus on the global dS2 spacetime, which can be
parametrized by $(t,\theta)$ coordinates with a metric given by the following
expression

$\qquad
ds^{2}=L^{2}\left(-dt^{2}+\cosh^{2}t\,d\theta^{2}\right)=\frac{L^{2}}{\cos^{2}T}\left(-dT^{2}+d\theta^{2}\right)\,.$
(2.1)

By applying the coordinate transformation $\sinh t=\tan T$, $\cosh
t=\frac{1}{\cos T}$ to the global coordinates, it is straightforward to obtain
the conformal coordinates with a compact Lorentzian time
$T\in(-\frac{\pi}{2},+\frac{\pi}{2})$. Given that the spatial direction in dS2
is one-dimensional, the spatial periodicity can be selected at will. For the
sake of simplicity, we will consider only the canonical case, wherein the
circumference of the spatial circle $S^{1}$ is equal to $2\pi L$, i.e.,
$\theta=[-\pi,\pi]$. As illustrated in the Penrose diagram for dS2, which is
shown in the right panel of figure 2, our choice corresponds to identifying
the two poles located at $\theta=\pm\pi$. For later purpose, we note that dS2
spacetime is conformally flat, i.e., the line element can be expressed as

$\quad ds^{2}=\frac{L^{2}}{\Omega^{2}}dzd\bar{z}=\frac{4L^{2}}{(1+z\
\bar{z})^{2}}dzd\bar{z}\,,$ (2.2)

where the conformal factor is given by $\Omega=\frac{1}{2}(1+z\bar{z})$. The
Lorentzian global coordinates $(z,\bar{z})$ are given as

$z=e^{-i(T-\theta)}\,,\qquad\bar{z}=e^{-i(T+\theta)}\,.$ (2.3)

In the following, we will assume that dS2 gravity is confined to a half
circle, e.g., with angular coordinate $\theta=[-\frac{\pi}{2},\frac{\pi}{2}]$,
where we impose the Neumann boundary condition. For the another half of the
circle, the gravity is turned off by imposing the Dirichlet boundary
condition. For the sake of convenience in calculations, it is assumed that the
dynamical matter on the full dS2 background is still CFT2. In other words, we
glue a half dS2 gravity with a CFT2 bath system along a defect line at
$\theta=\theta_{0}=\pm\frac{\pi}{2}$. See figure 3 for the illustration of the
system under consideration.

It should be noted that there are various gravitational theories that can
support a global dS2 solution. One of the simplest theories which can have the
dS2 solution is the Liouville gravity (or the 2d induced gravity):

$S_{\textrm{\tiny L}}=-\frac{c}{24\pi}\int
d^{2}x\left[(\partial_{T}\phi)^{2}-(\partial_{\theta}\phi)^{2}+\frac{e^{2\phi}}{L^{2}}\right]\,,$
(2.4)

where $\phi$ is the Liouville field such that the 2d metric reads
$ds^{2}=e^{2\phi}(-dT^{2}+d\theta^{2})$. This action is induced from a 2d CFT
with the central charge $c$ which is directly coupled with a dynamical metric,
without adding any action of 2d gravity. As argued in Suzuki:2022xwv  (see
also Chen:2020uac ; Chen:2020hmv ; Akal:2021foz ; Neuenfeld:2024gta ), this
theory is indeed realized as an effective theory on the EOW brane via the
brane-world type setup, which we will employ in section 3. It may also be
helpful to write down a covariant form of the Liouville gravity action as the
nonlocal Polyakov action Chen:2020uac , namely

$\displaystyle S_{\textrm{\tiny L}}=\frac{c}{96\pi}\int
dx^{2}\sqrt{-g}\left[-R\frac{1}{\Box}R-\frac{4}{L^{2}}\right].$ (2.5)

Moreover, distinct types of JT gravity can be obtained through the
implementation of different dimensional reductions from higher-dimensional
spacetime. For instance, one can obtain a dS JT gravity with positive
curvature coupled to a dynamical dilaton field, namely

$S_{\rm JT}=\frac{1}{16\pi G_{\textrm{\tiny N}}^{(2)}}\int
d^{2}x\sqrt{-g}\,\Phi\left(R-\frac{2}{L^{2}}\right)+\text{boundary terms}\,,$
(2.6)

by performing the spherical reduction from three-dimensional Einstein-Hilbert
action, see e.g., Sybesma:2020fxg ; Kames-King:2021etp ; Svesko:2022txo  for
more details. The dynamical dilaton field, denoted by $\Phi$, is naturally
interpreted as the radius in dS3. It is important to note that the dilaton
value in this theory must be constrained to be positive in order to ensure
that the effective Newton constant remains positive. Consequently, its dS2
solution encompasses the cosmological horizon, and its Penrose diagram is
identical to the global dS2 depicted in figure 2. Conversely, the dS version
of AdS JT gravity is derived by taking the limit of a higher-dimensional
extremal black hole Maldacena:2019cbz ; Cotler:2019nbi . Consequently, the
induced two-dimensional gravitational theory incorporates an additional
topological term, i.e.,

$S_{\rm JT^{\prime}}=\frac{\Phi_{0}}{16\pi G_{\textrm{\tiny N}}^{(2)}}\int
d^{2}x\sqrt{-g}R+\frac{1}{16\pi G_{\textrm{\tiny N}}^{(2)}}\int
d^{2}x\sqrt{-g}\,\Phi\left(R-\frac{2}{L^{2}}\right)+\text{boundary terms}\,,$
(2.7)

where the constant field $\Phi_{0}$ is related to the ground state entropy of
higher-dimensional extremal black hole and dynamical $\Phi$ controls the
deviation from the extremality. In comparison to previous dS2 gravity
theories, the dS JT gravity defined in eq. (A.1) allows for both cosmological
horizons and black holes due to the fact that the dilaton field $\Phi$ is
permitted to take negative values. As a result, the Penrose diagram for this
dS JT gravity is distinct from that of the global dS spacetime. In contrast,
it is analogous to a Schwarzschild-de Sitter spacetime.

In the remainder of this section, we will limit our focus to the most basic
case where two-dimensional gravity theory is induced from a 2d CFT, where the
integration of the CFT degrees of freedom lead to the Liouville action (2.4).
This setup is conveniently described by a classicial gravity in higher
dimensions via the double holography as will be studied in section 3. In this
case, there is no contribution to the entropy from the 2d gravity and the
entropy all comes from the 2d CFT sector. However, it is trivial to simply add
the topological term as in the first term in (A.1), which is proportional to
$\int d^{2}x\sqrt{-g}R$, if we like. This simply produces an additional
constant to the total gravitational entropy (or Bekenstein-Hawking entropy),
i.e.,

$\frac{\Phi_{0}}{4G_{\textrm{\tiny N}}^{(2)}}\equiv S_{0}=\text{constant}\,.$
(2.8)

The remaining two distinct types of JT gravity will be discussed in appendix
A. As a summary, it can be stated that all conclusions presented in the main
text also apply to the dS JT gravity with only cosmological horizon. However,
it would be subtle for dS JT gravity defined in eq. (A.1) due to the
appearance of the black hole singularity.

###  2.2 Island formula

Figure 4: Left: A time slice of our setup with dS gravity coupled to the CFT2
bath. The antipodal island is denoted by the fact that the endpoints of the
island are the antipodal points of the bath interval $\mathcal{A}$. Right: The
bath interval $\mathcal{A}$ and its corresponding antipodal island are shown
in the Penrose diagram.

We are interested in evaluating the entanglement entropy and its time
evolution for a boundary interval $\mathcal{A}$ situated within the CFT2 bath.
For the sake of simplicity, we will consider the symmetric interval defined by

$\mathcal{A}:\qquad\\{t=t_{A}\,,\quad\theta\in[-\pi,-\theta_{A}]\cup[\theta_{A},\pi]\\}\,.$
(2.9)

Drawing lessons from the breakthrough in understanding the fine-grained
entropy of Hawking radiation and the black hole information paradox, it is
pointed out that the correct formula for the fine-grained entropy of a non-
gravitating region entangled with a gravitational system is given by the so-
called island formula Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni
, i.e.,

$S_{\mathrm{EE}}(\hat{\rho}_{\mathcal{A}})=\underset{\mathcal{I}}{\operatorname{Min}}\left[\underset{\mathcal{I}}{\operatorname{Ext}}\left(\frac{\mathrm{Area}[\partial\mathcal{I}]}{4G_{\textrm{\tiny
N}}^{(2)}}+S_{\mathrm{matter}}\left(\mathcal{A}\cup\mathcal{I}\right)\right)\right]\,,$
(2.10)

where the first area term denotes the gravitational entropy 111It should be
noted that the corresponding expression for the “area” term is contingent upon
the gravitational theory under consideration. In a more general sense, it is
defined as the Wald-Dong entropy for a generic diffeomorphism-invariant
theory. and the second term, denoted as $S_{\mathrm{matter}}$, is often
referred to as the bulk entropy and represents the quantum corrections, which
are given by the von Neumann entropy of matter fields. The position of island
$\mathcal{I}$ is determined by its boundary $\partial\mathcal{I}$, i.e., the
quantum extremal surface (QES), which is obtained by extremizing the
generalized entropy $S_{\rm gen}$.

Although the original island formula was developed in AdS gravity coupled to a
bath, it is reasonable to expect that the same formula should also apply to dS
or flat gravity. A substantial body of prior research has also investigated
the island formula in dS gravity, with a particular focus on dS JT gravity,
e.g., Hartman:2020khs ; Sybesma:2020fxg ; Balasubramanian:2020xqf ; Aguilar-
Gutierrez:2021bns ; Kames-King:2021etp ; Svesko:2022txo ; Aalsma:2022swk . In
the following section, we will attempt to calculate the fine-grained entropy
of the boundary interval $\mathcal{A}$, by directly applying the island
formula. Nevertheless, it will become evident that the subsequent naive island
solution presents inherent issues from different perspectives.

####  2.2.1 Naive antipodal island in dS2 gravity

Figure 5: The phase transition between the entropy $S_{\textrm{\tiny
EE}}(\hat{\rho}_{\textrm{\tiny A}})$ for a bath interval $\mathcal{A}$ in the
non-island phase and that derived by the island formula. It gives the same
results as the standard holographic entanglement entropy $S_{\mathcal{A}}$. We
set $\theta_{A_{1}}-\theta_{A_{2}}=\frac{\pi}{4}$, $\epsilon=0.01$ for this
plot.

In light of the assumption that gravitational entropy is merely a constant
(eq. (2.8)), the remaining objective for evaluating the island formula is to
calculate the bulk entropy, or the von Neumann entropy of intervals in CFT2.
Noting the fact the CFT2 matter is living on global dS2 space. To derive the
entanglement entropy of a single interval $[z_{1},z_{2}]$ in the global dS
coordinates, we can use the flat-spacetime result with counting the
appropriate Weyl factors for each endpoints of the interval. By employing the
Weyl factor specified in eq. (2.2), it is straightforward to get

$\begin{split}S_{\rm
matter}\left([\theta_{1},\theta_{2}]\right)&=\frac{c}{6}\log\left(\frac{(z_{1}-z_{2})(\bar{z}_{1}-\bar{z}_{2})}{\epsilon_{1}\epsilon_{2}\Omega_{1}\Omega_{2}}\right)\\\
&=\frac{c}{6}\log\left(\frac{2\left(\cos(T_{1}-T_{2})-\cos(\theta_{1}-\theta_{2})\right)}{\epsilon_{1}\epsilon_{2}\cos
T_{1}\cos T_{2}}\right)\\\ &=\frac{c}{6}\log\left(\frac{2(1+\sinh t_{1}\sinh
t_{2}-\cosh t_{1}\cosh
t_{2}\cos(\theta_{1}-\theta_{2}))}{\epsilon_{1}\epsilon_{2}}\right)\,,\end{split}$
(2.11)

where we have introduced two UV cut-off $\epsilon_{i}$ associated with two
individual endpoints. Note that we are always working on the Hartle-Hawking
state. Here we have specified the von Neumann entropy in terms of the global
Lorentzian time $t$, which is derived from the Euclidean signature by Wick
rotation. In the next section we will see that the same answer can be obtained
by evaluating the geodesic length in AdS3.

In the non-island phase, in the absence of contributions from the island, the
entanglement entropy of a single interval $\mathcal{A}$ is given by the
following expression:

$S_{\mathrm{EE}}(\hat{\rho}_{\mathcal{A}})\big{|}_{\rm{non-island}}=S_{\rm
matter}\left(\mathcal{A}\right)=\frac{c}{3}\log\left(\frac{2\cosh
t_{A}\cdot\sin\theta_{A}}{\epsilon}\right)\,,$ (2.12)

which presents a linear growth

$S_{\mathrm{EE}}(\hat{\rho}_{\mathcal{A}})\big{|}_{\rm{non-
island}}\simeq\frac{c}{3}t_{A}\,.$ (2.13)

at late times. This can be understood as a reflection of the inflation of the
dS2 background.

Furthermore, let us consider the island phase with the assumption that the
boundary of the island region is located at

$\partial\mathcal{I}:\qquad\left(t=t_{I},\theta=\pm\theta_{I}\right)\,.$
(2.14)

Since the global Hartle-Hawking state is pure, the bulk entropy in the island
phase can be calculated by considering the complimentary region of
$\mathcal{A}\cup\mathcal{I}$, which is the two-interval region
$(\mathcal{A}\cup\mathcal{I})^{c}$. This calculation for bulk entropy
corresponds to evaluating two two-point functions associated with endpoints at
$\theta=\theta_{A}$ and $\theta=\theta_{I}$. The corresponding generalized
entropy at the island phase thus can be obtained in a straightforward manner
by applying the two-point function derived in eq. (2.11), i.e.,

$\begin{split}S_{\rm gen}&=S_{0}+S_{\rm
matter}\left([\theta_{I},\theta_{A}]\right)+S_{\rm
matter}\left([-\theta_{A},-\theta_{I}]\right)\\\
&=S_{0}+\frac{c}{3}\log\left(\frac{2(1+\sinh t_{I}\sinh t_{A}-\cosh t_{I}\cosh
t_{A}\cos(\theta_{I}-\theta_{A}))}{\epsilon\,\epsilon_{I}}\right)\,,\\\
&=\tilde{S}_{0}+\frac{c}{3}\log\left(\frac{2(1+\sinh t_{I}\sinh t_{A}-\cosh
t_{I}\cosh t_{A}\cos(\theta_{I}-\theta_{A}))}{\epsilon}\right)\,,\end{split}$
(2.15)

where the cutoff scale at the island region, $\epsilon_{I}$, has been absorbed
into the Newton constant $S_{0}$, by introducing
$\tilde{S}_{0}=S_{0}+\frac{c}{3}\log\frac{1}{\epsilon}$. The extremization
with respect to the spatial direction $\theta_{I}$ and the timelike direction
$t_{I}$ is simply given by solving the following equations:

$\begin{split}\partial_{\theta_{I}}S_{\rm gen}=0&\longrightarrow\quad\cosh
t_{I}\cosh t_{A}\sin(\theta_{I}-\theta_{A})=0\,,\\\ \partial_{t_{I}}S_{\rm
gen}=0&\longrightarrow\quad\cosh t_{I}\sinh t_{A}-\sinh t_{I}\cosh
t_{A}\cos(\theta_{I}-\theta_{A})=0\,.\end{split}$ (2.16)

It is evident that our choice (2.8) facilitates the extremization over the
gravitational region, as the derivative of the gravitational entropy vanishes.
The solution for the extremization is unique and is given by the following
equation:

$\theta_{I}=-\pi+\theta_{A}\,,\quad t_{I}=-t_{A}\,,$ (2.17)

which will be called the antipodal island, as illustrated in figure 4.

The distinguishing feature related to this type island is that it moves back
in time as the bath interval involves in time. The trivial solution
$\theta_{I}=\theta_{A},t_{I}=t_{A}$ is naturally discarded because it is
outside the gravitational region. Consequently, the island formula associated
with this antipodal island gives rise to a constant, i.e.,

$S_{\mathrm{EE}}(\hat{\rho}_{\mathcal{A}})\big{|}_{\rm{island}}=S_{0}+\frac{2c}{3}\log\frac{2}{\epsilon}=\tilde{S}_{0}+\frac{c}{3}\log\left(\frac{4}{\epsilon}\right)\,,$
(2.18)

which would be always smaller than that from the non-island phase at late
times. The phase transition from the non-island phase to the island phase is
thus given by

$t_{\ast}=\text{arccosh}\left(\frac{2\exp(\frac{3}{c}\tilde{S}_{0})}{\sin\theta_{A}}\right)\approx\frac{3}{c}\tilde{S}_{0}-\log\left(\frac{\sin\theta_{A}}{4}\right)\,.$
(2.19)

The time evolution of the entanglement entropy, as calculated by the island
rule, is depicted in figure 5. Similar types of island in JT gravity has also
been discussed in detail in Hartman:2020khs ; Sybesma:2020fxg ; Aguilar-
Gutierrez:2021bns ; Kames-King:2021etp ; Svesko:2022txo .

####  2.2.2 Problems with antipodal islands

As indicated in the title of this section, we will demonstrate that the naive
antipodal island is not physical for various reasons. The attentive reader may
have noticed that the previously identified antipodal island introduces
ambiguity in defining the island region. As illustrated in figure 4, we define
the island region by fixing its two endpoints, i.e.,

$\partial\mathcal{I}:\qquad\left(-t_{A}\,,-\theta_{A}+\pi\right)\cup\left(-t_{A}\,,\theta_{A}-\pi\right)\,.$
(2.20)

However, it is important to note that the calculations for the matter entropy
part are based on the calculations of the two-point functions associated with
the two-interval subregion, which is defined as
$(\mathcal{A}\cup\mathcal{I})^{c}$. Substitution of the island solution
derived by extremization, namely the antipodal island, reveals that the time
slice consisting of
$(\mathcal{A}\cup\mathcal{I})^{c}\cup(\mathcal{A}\cup\mathcal{I})$ becomes
ill-defined. Noticing the fact222According to our convention, the endpoint
located at $\theta_{A}$ in the bath region is constrained by
$\theta_{A}\in[\pi/2,\pi]$.

$-\theta_{A}+\pi\geq\theta_{A}-\pi\,,$ (2.21)

It is evident that the corresponding time slice encompassing the bath interval
and island region is counted three times, which is an obvious discrepancy.
From this perspective, it can be concluded that the antipodal island cannot be
considered a valid or physical solution, despite being the unique solution for
extremalizing the generalized entropy. As will be seen in the following
section, the entanglement wedge associated with the antipodal island is not
well-defined when the so-called double holography is employed.

One might inquire as to the origin of the issue associated with antipodal
islands. This inquiry may lead to the conclusion that the gravitational
entropy is taken as a constant, which is a simple choice. Nevertheless, it is
necessary to demonstrate that the antipodal island is invalid from a physical
standpoint. This is because the quantum extremal surface may not always be the
dominant saddle in calculations of fine-grained entropy. It is important to
recall that the quantum extremal surface is a generalization of the RT
surface, which is defined as the maximin surface in the bulk spacetime with
the Lorentzian signature. It is reasonable to posit that the quantum extremal
surface is also given by a maximin surface which extremizes the generalized
entropy $S_{\rm gen}$. However, the antipodal island, which resides in the dS
gravitational region, is instead maximal in the spatial direction and minimal
in the timelike direction. To demonstrate this assertion, we may examine the
second derivative of the generalized entropy, $S_{\rm gen}$, with respect to
the location of islands. More explicitly, it is straightforward to get

$\begin{split}\frac{\partial^{2}S_{\rm
gen}}{\partial\theta_{I}\partial\theta_{I}}\bigg{|}_{t_{I}=-t_{A},\theta_{I}=\theta_{A}-\pi}&=-\frac{1}{2}\cosh^{2}\theta_{A}<0\,,\\\
\frac{\partial^{2}S_{\rm gen}}{\partial t_{I}\partial
t_{I}}\bigg{|}_{t_{I}=-t_{A},\theta_{I}=\theta_{A}-\pi}&=\frac{1}{2}>0\,.\\\
\end{split}$ (2.22)

The preceding two inequalities indicate that the antipodal island, as
perceived from the perspective of the generalized entropy, is maximal in the
spatial direction and minimal in the timelike direction, which is opposite to
the property of RT surface for holographic entanglement entropy. In the
Euclidean signature, the antipodal island thus corresponds to the maximum in
the Euclidean time direction. Although the quantum extremal surface related to
the antipodal island in dS gravity corresponds to a saddle point, it can be
concluded that it cannot dominate the calculation for the fine-grained entropy
due to the absence of the maximin surface. Regardless of the properties of the
gravitational or non-gravitational region, it can be expected that this is a
common problem: the quantum extremal surface corresponding to the dominated
saddle does not exist. To address this issue, it is necessary to develop a
novel approach for calculating entanglement entropy that does not rely on the
island formula. In the following section, we seek to identify a solution by
moving to a higher dimension. To this end, we employ the double holography in
AdS3, which reveals that the non-extremal island plays a important role in the
semi-classical limit.

##  3 Non-extremal island from double holography

In this section, we explore a doubly holographic setup involving a dS2
braneworld coupled to a non-gravitational bath. This doubly holographic
perspective provides additional evidence that the naive island formula cannot
be directly applied in the de Sitter braneworld, as discussed in the previous
section. Leveraging higher-dimensional holography, the entanglement entropy of
a bath interval can be accurately computed using the holographic correlation
function in the AdS/BCFT correspondence. We further show that the appropriate
formula for evaluating the entanglement entropy associated with dS gravity is
given by the non-extremal island situated at the edge of the dS gravitational
region.

###  3.1 Double holography with a dS braneworld

Figure 6: Three perspectives in double holography for dS gravity coupled to a
non-gravitational bath residing in dS spacetime. The image in the middle shows
the brane perspective, which is the same as shown in figure 3 in section 2.

We begin by constructing a doubly holographic model for the setup discussed in
the previous section, specifically dS2 gravity coupled to a CFT2 bath. The
essential concept entails the embedding of the dS2 brane within an AdS3 bulk
spacetime, utilizing the AdS3/dS2 slicing. This can be expressed as follows:

$ds^{2}=L_{\textrm{\tiny
AdS}}^{2}\quantity(d\eta^{2}+\sinh^{2}{\eta}\quantity(-dt^{2}+\cosh^{2}{t}\;d\theta^{2}))\,,$
(3.1)

where $L_{\textrm{\tiny AdS}}$ denotes the AdS radius and each
$\eta=\mathrm{const}>0$ surface is nothing but a dS2 spacetime. The Euclidean
AdS3 spacetime with the Euclidean dS brane is obtained by performing the Wick
rotation $t=-i{t_{\textrm{\tiny E}}}$:

$ds^{2}=L_{\textrm{\tiny
AdS}}^{2}\quantity(d\eta^{2}+\sinh^{2}{\eta}\quantity(d{t_{\textrm{\tiny
E}}}^{2}+\cos^{2}{{t_{\textrm{\tiny E}}}}\;d\theta^{2})).$ (3.2)

The Euclidean time ${t_{\textrm{\tiny E}}}$ runs in the range
${t_{\textrm{\tiny E}}}\in[-\frac{\pi}{2},\frac{\pi}{2}]$. The asymptotic
boundary of AdS3 can be approached by taking the limit $\eta\to\infty$. In the
following, we assume that the UV cut-off surface is located at
$e^{\eta}=\coth\left(\frac{\epsilon}{2}\right)\approx\frac{2}{\epsilon}$ with
identifying $\epsilon\,L_{\textrm{\tiny AdS}}$ as the UV cutoff in the CFT2.

To align with the previous setup involving dS2 gravity and a CFT$2$ bath, we
cut the global AdS3 by a constant $\eta$ slice and assume that the boundary of
the bulk spacetime is located at $\eta=\eta_{\infty}$, see figure 6 for an
illustration. Since only part of the dS$2$ brane is gravitational, we need to
impose inhomogeneous boundary conditions on the dS2 brane located at
$\eta=\eta_{\infty}$:

  1. 1.

For $-\theta_{D}\leq\theta\leq\theta_{D}$, we identify this region as an End-
of-World brane and thus impose the Neumann boundary condition, i.e.,

$K[g]_{ij}-K[g]g_{ij}+\mathcal{T}g_{ij}=0\,.$ (3.3)

where $g_{ij}$ and $K[g]_{ij}$ denote the induced metric and extrinsic
curvature of the brane, respectively. It is worth noting that a dS brane is
only supported as a solution when its tension $\mathcal{T}$ satisfies
$|\mathcal{T}|>1/L{\textrm{\tiny AdS}}$. More explicitly, the tension of the
brane determines its location in terms of
$\mathcal{T}=\frac{\coth\eta_{\infty}}{L_{\textrm{\tiny AdS}}}$.333Note that
the sign of the tension, as well as the extrinsic curvature, will play an
important role in the analysis in section 4. The trace of the extrinsic
curvature of the dS brane, i.e., , $K$, is positive with respect to the
physical spacetime defined by the bulk region enclosed by the dS brane.

  2. 2.

For $\theta_{D}\leq\theta\leq\pi$ and $-\pi\leq\theta\leq-\theta_{D}$, we
assign the Dirichlet boundary condition as the normal boundary condition for
the AdS/CFT correspondence. For simplicity, we can set
$\theta_{D}=\frac{\pi}{2}$ in the following calculations. Since we expect that
the field theory living on the dS brane is a holographic CFT2, this region
should be placed at the conformal boundary. This can be achieved by taking the
large $\eta$ limit, i.e., relating $\eta\infty$ to the (dimensionless) UV cut-
off $\epsilon$ as follows:

$\eta_{\infty}\approx\log\left(\frac{2}{\epsilon}\right)\,.$ (3.4)

This AdS3 bulk spacetime with an EOW dS2 brane serves as the holographic dual
of the model discussed in the previous section. Furthermore, we argue that
this is, in fact, a doubly holographic model. In other words, there are three
equivalent dual descriptions from different perspectives at different
dimensions:

  1. (A)

Bulk perspective: AdS3 bulk spacetime with an EOW dS2 brane. We have the
standard Einstein gravity living in AdS3 spacetime with a dS2 boundary, where
the Dirichlet (and Neumann) boundary condition is imposed on one half of dS2
(and the other half dS2). The latter half dS2 with the Neumann boundary
condition can be regarded as an EOW brane, as shown in figure 6.

  2. (B)

Brane perspective: 2d induced gravity on a dS2 braneworld is coupled to the 2d
non-gravitational field theory living on dS2 spacetime. We have a CFT on one
half of dS2 and induced gravity (or equivalently Liouville gravity) on the
other half of dS2, which are joined together along the timelike boundary. See
the middle panel of figure 6.

  3. (C)

Boundary perspective: BCFT2 living in a half dS2 spacetime. There is a defect
line located at $\theta=\theta_{D}$. This one-dimensional defect can be
interpreted as the holographic dual of a half dS2 gravitational spacetime
Kawamoto:2023nki .

The equivalence between (A) and (B) is elucidated by braneworld holography
Randall:1999ee ; Randall:1999vf ; Gubser:1999vj ; Karch:2000ct  in the context
of the dS brane. The induced gravity on the dS2 brane, specifically the
Liouville gravity (2.4), can be derived by performing dimensional reduction
along the radial direction Suzuki:2022xwv ; Kawamoto:2023wzj ; Akal:2020wfl ;
Neuenfeld:2024gta . It is important to note that the reduced theory on dS2
also includes a CFT matter sector, where divergences in the matter sector
precisely cancel those appearing in the gravitational action. The duality
between (B) and (C) can be supported by the half dS holography proposed in
Kawamoto:2023nki , which posits that the gravitational theory in a half dSd+1
spacetime with a Dirichlet boundary is dual to a non-local quantum field
theory 444Although it is not immediately clear that the boundary dual theory
is non-local, our findings regarding the non-extremal island (defined in later
subsections) provide an alternative demonstration of this non-locality.
Specifically, the island corresponding to a finite bath interval is observed
to be spread across the entire gravitational region, rather than being
confined by the quantum extremal surface. residing on this $d$-dimensional
timelike boundary. A qualitatively similar yet distinct holographic setup
involving dS bulk duals is constructed through $T\overline{T}$ deformation
Alishahiha:2004md ; Dong:2018cuv ; Gorbenko:2018oov . We will not delve into
this dS holography in this paper, as it is not critical for the calculations
presented herein. Finally, the holographic duality between (A) and (C) follows
from the AdS/BCFT correspondence Takayanagi:2011zk ; Fujita:2011fp ;
Karch:2000gx , where the dS brane setup was first considered in Akal:2020wfl .
Additionally, refer to the recent paper Wei:2024zez  for an interesting
application of dS branes to cross-cap states. In this paper, we will focus on
the holographic duality between (A) and (B).

The construction of this doubly holographic model is summarized in figure 6.
The advantage of this approach is that the bulk perspective, which employs a
higher-dimensional AdS spacetime, offers an alternative method for calculating
entanglement entropy. This aids in resolving the puzzles associated with the
island formula in de Sitter gravity.

###  3.2 Naive holographic entanglement entropy is problematic

Figure 7: Left: A schematic diagram of the disconnected antipodal RT surface
in AdS3. The two antipodal RT surfaces (red lines) always intersect on the
$t=0$ surface. Right: Sketch of a time slice in our setup. The red region
denotes the EOW brane, while the green region represents the CFT bath with
Dirichlet boundary conditions. The subsystem $\mathcal{A}$, for which we
calculate the entanglement entropy, is indicated by the blue curve. The red
lines in the bulk denote the RT surfaces for the disconnected phase, and the
purple curve represents the RT surface for the connected phase.

Similar to the previous section, we will focus on calculating the entanglement
entropy of an interval $\mathcal{A}$ in the non-gravitational region. Without
loss of generality, let us consider a subsystem $\mathcal{A}$ at a constant
time slice $t=t_{A}$ on the dS2 brane given by $\eta=\eta_{\infty}$ and choose
a finite interval $\mathcal{A}=[\theta_{A_{1}},\theta_{A_{2}}]$ within the
Dirichlet region555For a symmetric interval centered in the middle of the
space, as shown in figure 4, one would choose
$A=[\theta_{A},\pi]\cup[-\pi,-\theta_{A}]$.. Leveraging the holographic
duality with the AdS3 bulk spacetime, we can apply the holographic
entanglement entropy (HEE) formula Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt ;
Faulkner:2013ana ; Engelhardt:2014gca  to compute the entanglement entropy of
the subsystem $\mathcal{A}$ on the conformal boundary. We will explicitly
demonstrate that this approach yields the same results as the island formula.
However, we will also find that the HEE approach does not provide physically
accurate results in this context.

####  3.2.1 Holographic entanglement entropy

Since there is an EOW brane in the bulk AdS3 spacetime, there are two types of
extremal surfaces according to the prescription of the AdS/BCFT correspondence
Takayanagi:2011zk ; Fujita:2011fp ; Akal:2021foz ; Akal:2022qei . The first
candidate is the connected extremal surface denoted by
$\Gamma^{\mathrm{con}}_{\mathcal{A}}$, whose boundary is located only on the
entangling surface $\partial\mathcal{A}$. The second candidate is the
disconnected surface $\Gamma^{\mathrm{dis}}_{\mathcal{A}}$, which is anchored
on the EOW brane. We denote the corresponding contributions to the
entanglement entropy as $S_{\mathcal{A}}^{\mathrm{con}}$ and
$S_{\mathcal{A}}^{\mathrm{dis}}$, respectively. Applying the holographic
entanglement entropy (HEE) formula, the entanglement entropy of the bath
interval $\mathcal{A}$ is given by

$S_{\mathcal{A}}=\min\\{S_{\mathcal{A}}^{\mathrm{con}},S_{\mathcal{A}}^{\mathrm{dis}}\\}=\min\left\\{\frac{\text{Area}(\Gamma_{\mathcal{A}}^{\rm{con}})}{4G_{\textrm{\tiny
N}}},\frac{\text{Area}(\Gamma_{\mathcal{A}}^{\rm{dis}})}{4G_{\textrm{\tiny
N}}}\right\\}\,,$ (3.5)

where $G_{\textrm{\tiny N}}$ denotes the Newton’s constant for the
gravitational theory in the AdS3 bulk spacetime and is distinct from the one
for the induced gravity on the dS brane. To evaluate the area of extremal
surfaces in AdS3, we only need the dimensionless length of the spacelike
geodesic $D_{ab}$, defined as:

$\cosh D_{ab}=\cosh\eta_{a}\cosh\eta_{b}-\sinh\eta_{a}\sinh\eta_{b}\left(\cosh
t_{a}\cosh t_{b}\cos\left(\theta_{b}-\theta_{a}\right)-\sinh t_{a}\sinh
t_{b}\right)\,.$ (3.6)

where the spacelike geodesic connects two points $(t_{a},\eta_{a},\theta_{a})$
and $(t_{b},\eta_{b},\theta_{b})$ in AdS3 spacetime.

Connected Extremal Surface $\Gamma_{\mathcal{A}}^{\rm{con}}$: Considering the
geodesics connecting the two endpoints of the interval $\mathcal{A}$, namely:
$A_{1}=(\eta_{\infty},t_{A},\theta_{A_{1}})$ and
$A_{2}=(\eta_{\infty},t_{A},\theta_{A_{2}})$, the corresponding entropy
contribution is given by

$S_{\mathcal{A}}^{\mathrm{con}}=\frac{L_{\textrm{\tiny
AdS}}D_{12}}{4G_{N}}=\frac{c}{3}\log\cosh{t_{A}}+\frac{c}{3}\log\quantity(\frac{2}{\epsilon}\absolutevalue{\sin\quantity(\frac{\theta_{A_{1}}-\theta_{A_{2}}}{2})})\,.$
(3.7)

Here, we use the Brown-Henneaux formula Brown:1986nw :

$c=\frac{2L_{\mathrm{AdS}}}{3G_{N}}\,.$ (3.8)

The linear growth along the global time $t_{\mathcal{A}}$ reflects the
exponential inflation of the dS background. This holographic formula from AdS3
agrees with the result derived from the brane perspective in the non-island
phase, as indicated by eq. (2.12), after identifying the size of the interval
$\mathcal{A}$.

Disconnected Extremal Surface $\Gamma_{\mathcal{A}}^{\rm dis}$: In this phase,
the extremal surfaces are disconnected, with each surface connecting the
boundary of the interval $\mathcal{A}$ to the EOW brane. For a given extremal
surface, we denote the endpoints on the AdS3 boundary by
$A_{i}=(\eta_{\infty},t_{A_{i}},\theta_{A_{i}})$ and the endpoints on the EOW
brane by $I_{i}=(\eta_{\infty},t_{I_{i}},\theta_{I_{i}})$. Focusing on one
geodesic, we simplify by omitting the subscript $i=1,2$. The location of
$I_{i}$ can be fixed by solving the extremization conditions, namely

$\frac{\partial D_{AI}}{\partial t_{I}}=0,\;\frac{\partial
D_{AI}}{\partial\theta_{I}}=0.$ (3.9)

Using the universal formula (3.6) for the geodesic length, its derivatives are

$\begin{split}\sinh{D_{AI}}\frac{\partial
D_{AI}}{\partial\theta_{I}}&=\sinh^{2}{\eta_{\infty}}\cosh{t_{A}}\cosh{t_{I}}\sin{(\theta_{I}-\theta_{A})},,\
\\\ \sinh{D_{AI}}\frac{\partial D_{AI}}{\partial
t_{I}}&=\sinh^{2}{\eta_{\infty}}\left(\cosh{t_{A}}\sinh{t_{I}}\cos(\theta_{A}-\theta_{I})-\sinh{t_{A}}\cosh{t_{I}}\right)\,.\end{split}$
(3.10)

Constrained to the EOW brane666In this computation, we do not need to specify
the size of the Dirichlet region, i.e., the value of $\theta_{D}$. However, if
the Dirichlet region is larger than half the size of the entire space, there
is no disconnected RT surface. We will assume that EOW brane on the spatial
circle is bigger enough such that the disconnected extremal surface could
exist., the extremal conditions yield a single solution:

$\theta_{I}=\theta_{A}-\pi\,,\qquad t_{I}=-t_{A}\,.$ (3.11)

As illustrated in figure 7, the extremal disconnected RT surface always
connects antipodal points on the spatial circle. This location on the EOW
brane corresponds precisely to the quantum extremal surface, i.e., the
boundary of the antipodal island (2.17) derived from the island formula.

One can find that the holographic entanglement entropy associated with the
extremal geodesic $\Gamma_{\mathcal{A}}^{\rm dis}$ reduces to a constant as
follows

$\begin{split}S_{\mathcal{A}}^{\mathrm{dis}}=\frac{L_{\textrm{\tiny
AdS}}}{4G_{\textrm{\tiny
N}}}\left(D_{A_{1}I_{1}}+D_{A_{2}I_{2}}\right)=\frac{\eta_{\infty}}{G_{\textrm{\tiny
N}}}&=\frac{2c}{3}\eta_{\infty}\approx\frac{2c}{3}\log\left(\frac{2}{\epsilon}\right)\,,\end{split}$
(3.12)

After choosing $S_{0}=0$,777This is due to the fact that all contributions on
the brane are assigned as originating from the CFT2 (Liouville field) that
resides on the EOW brane. An alternative approach would be to consider all
contributions on the EOW as originating from the effective gravity on the dS2
brane and rewrite $S_{\mathcal{A}}^{\mathrm{dis}}$ as
$S_{\mathcal{A}}^{\mathrm{dis}}=2\times\left(\frac{\eta_{\infty}}{4G_{\textrm{\tiny
N}}}+\frac{c}{6}\log\left(\frac{2}{\epsilon}\right)\right)\,,$ (3.13) where
the effective Newton’s constant on dS2 brane is given by
$\frac{1}{G_{\textrm{\tiny N}}^{(2)}}=\frac{\eta_{\infty}}{G_{\textrm{\tiny
N}}}$. this reproduces our previous result (2.18) that is derived from the
direct application of the island formula from the brane perspective. From the
equivalent boundary perspective, we can also recast this result in terms of

$S_{\mathcal{A}}^{\mathrm{dis}}=2\times\left(\frac{c}{6}\log\left(\frac{2}{\epsilon}\right)+S_{\rm
bdy}\right)\,,$ (3.14)

where the boundary entropy is defined as $S_{\rm
bdy}=\frac{c}{6}\log\sqrt{\frac{\mathcal{T}+L_{\textrm{\tiny
AdS}}}{\mathcal{T}-L_{\textrm{\tiny AdS}}}}=\frac{c}{6}\eta_{\infty}$.

####  3.2.2 Problems with the disconnected extremal surface

The resulting HEE for an interval $\mathcal{A}$ is determined by the minimal
value between $S_{\mathcal{A}}^{\rm con}$ (3.7) and $S_{\mathcal{A}}^{\rm
dis}$ (3.12). As illustrated by figure 5, the connected extremal surface
$\Gamma_{\mathcal{A}}^{\rm con}$ is favored at early times, leading to a HEE
that exhibits linear growth with time. This growth can be described by eq.
(3.7). After the critical time $t_{\ast}$, given by

$t_{\ast}\approx\log\left(\frac{4}{\epsilon}\right)-\log\left(\absolutevalue{\sin\quantity(\frac{\theta_{A_{1}}-\theta_{A_{2}}}{2})}\right)\,,$
(3.15)

a phase transition occurs between the connected extremal surface
$\Gamma_{\mathcal{A}}^{\rm con}$ and the disconnected extremal surface
$\Gamma_{\mathcal{A}}^{\rm dis}$. After this transition, the HEE reaches a
constant value derived in eq. (3.12).

Nevertheless, we argue that the holographic entanglement entropy calculation
based on the conventional RT formula is not physical. Specifically, the
disconnected extremal surface connecting two antipodal points on the dS space
presents issues and appears unphysical. As discussed in the previous section,
similar problems arise with the antipodal island. The application of double
holography offers further evidence from the perspective of the bulk spacetime.

First and foremost, it is important to emphasize that the two disconnected
extremal RT surfaces always intersect at the center of the bulk spacetime,
specifically at $t=0,\theta=0$, as shown in figure 7. This intersection
indicates that these extremal surfaces fail to meet the general conditions for
RT surfaces and cannot be considered physical. Furthermore, this intersection
makes the entanglement wedge Czech:2012bh ; Wall:2012uf ; Headrick:2014cta
ill-defined, as illustrated in the right panel of figure 7. On the other hand,
the extremal geodesics anchored on the EOW brane represent the minimax surface
along the Lorentzian dS brane. It is easy to get the second derivatives of the
geodesic length at the antipodal solution. To wit,

$\begin{split}\left.\frac{\partial^{2}D_{AI}}{\partial
t_{I}^{2}}\right|_{\mathrm{antipodal}}&=-\left.\frac{\partial^{2}D_{AI}}{\partial({t_{\textrm{\tiny
E}}})_{I}^{2}}\right|_{\mathrm{antipodal}}=\frac{1}{2}\tanh{\eta_{\infty}}>0,\\\
\left.\frac{\partial^{2}D_{AI}}{\partial\theta_{I}^{2}}\right|_{\mathrm{antipodal}}&=-\frac{1}{2}\cosh^{2}{t_{0}}\tanh{\eta_{\infty}}<0\,.\end{split}$
(3.16)

These calculations show that a disconnected RT surface that is extremized
along the dS brane is maximal with respect to variations along both the
spatial and Euclidean time directions. Therefore, it is reasonable to infer
that this type of extremal surface does not correspond to the dominant saddle
in the Euclidean path integral for calculating the entanglement entropy. This
behavior is particularly pathological when compared with the holographic
entanglement entropy for an AdS brane. For a detailed comparison with the AdS
brane scenario, please refer to appendix B.

Given these issues, we conclude that the standard holographic entanglement
entropy formula and the island formula are not applicable to dS EOW branes or
dS gravity coupled to a non-gravitational bath.

####  3.2.3 Improved doubly holographic model

dS2 EOW branedS2 DirichletAdS2 Dirichlet braneAdS2 Dirichlet braneAdS3
bulk$\eta=\eta_{b}$$\eta=\eta_{\infty}$

dS2AdS2dS2AdS2dS2CFT2GravityCFT2

Figure 8: Sketches of the doubly holographic setup with a dS brane at finite
radius. The left panel shows the AdS3 bulk spacetime with an EOW brane:
Dirichlet boundary conditions are imposed on the large-radius dS2 cut (green)
and the AdS2 cut (orange), while Neumann boundary conditions are imposed on
the finite-radius dS brane (red). The dotted line indicates the conformal
boundary of the global AdS spacetime. The right panel shows the picture from
the brane perspective: there is no gravity in the green and orange regions,
while induced gravity is present in the red region.

Before proceeding, we would like to address an additional UV divergence that
appears in the doubly holographic model with a dS EOW brane located at
$\eta=\eta_{\infty}$. To join the EOW brane to the Dirichlet brane near the
conformal boundary, we must choose
$\eta_{\infty}\approx\log\left(\frac{2}{\epsilon}\right)$. Consequently, the
contributions from the area of the disconnected RT surface will always contain
twice the UV divergence compared to the connected ones. One might argue that
these issues associated with the disconnected RT surface anchored to the dS
brane are not problematic, as it would take an infinite time $t_{\ast}$ (3.53)
in the limit $\epsilon\to 0$ for the disconnected configuration to be favored.

Nevertheless, we posit that this phenomenon is merely a consequence of the
intrinsic simplicity of the dS2 brane solution within this doubly holographic
model. To avoid this confusion, it may be beneficial to consider a more
physical, but somewhat more complicated brane configuration. As shown in
figure 8, we consider an EOW brane with a finite dS radius, choosing
$\eta=\eta_{b}\ll\eta_{\infty}$ and imposing the Neumann boundary condition.
In the CFT region at the conformal boundary with the Dirichlet boundary
condition, we fix it as a dS2 brane with
$\eta=\eta_{\infty}\sim\log\left(\frac{2}{\epsilon}\right)$. To connect the
two dS regions with different radii, it is necessary to include an AdS2 region
with the Dirichlet boundary condition. In the symmetric case with
$\theta_{D}=\frac{\pi}{2}$, the intermediate AdS2 region corresponds to a
brane with vanishing tension. From the two-dimensional brane perspective, this
doubly holographic model can be viewed as dS2 gravity coupled to a non-
gravitational bath with a background that includes both a half dS2 and a
finite AdS2 spacetime. By taking the limit $\eta_{b}\to\eta\infty$, we can
recover the previous model where the AdS2 region shrinks to nothing. In the
next section, we will focus on the EOW dS2 brane with a finite $\eta_{b}$ and
demonstrate how the bulk holography in AdS3 can help to solve the puzzles
associated with the antipodal island and the disconnected RT surfaces.

###  3.3 Non-extremal Island on Euclidean dS brane

Although we have argued that the standard holographic entanglement formula,
which evaluates the area of extremal surfaces, cannot provide the correct
answer, we can still calculate the entanglement entropy $S_{\mathcal{A}}$ by
taking the advantage of holography. Specifically, in the setting of a
subsystem $\mathcal{A}$ in the non-gravitational half dS2 region coupled to a
gravity theory on another half dS2, with or without an intermediate AdS2
region, the entanglement entropy can be calculated using the replica method
Calabrese:2004eu ; Calabrese:2009qy . By taking the von Neumann limit with
$n\to 1$, the entanglement entropy can be derived as the two-point function of
twist operators at the endpoints, namely

$S_{\mathcal{A}}\equiv\lim_{n\to
1}\frac{1}{1-n}\log\langle\sigma_{n}(t_{A_{1}},\theta_{A_{1}})\bar{\sigma}_{n}(t_{A_{2}},\theta_{A_{2}})\rangle\,,$
(3.17)

where the normalization of twist operators is fixed as
$\langle\sigma_{1}(t_{A_{1}},\theta_{A_{1}})\bar{\sigma}_{1}(t_{A_{2}},\theta_{A_{2}})\rangle=1$
and their conformal dimension is $\Delta_{n}=\frac{c}{12}(n-\frac{1}{n})$. In
the semi-classical limit of a holographic field theory, this is equivalent to
taking a large central charge limit $\Delta_{n}\sim
c\sim\frac{1}{G_{\textrm{\tiny N}}}\to\infty$, which allows for a bulk dual
description 888A meticulous reader may have noticed that the two limits,
$c\to\infty$ and $n\to 1$, are mutually exclusive from the point of view of
the dimensionless parameter $\Delta_{n}$. This ambiguity could be circumvented
by carefully dealing with the order of the limits. We first take $c\to\infty$
limit to compute the holographic correlation function
$\frac{\log\expectationvalue{\sigma_{n}}}{\Delta_{n}}$ and then use $n\to 1$
to get the von Neumann entropy.. For example, the two-point function of CFT2
with a large charge can be derived by using the geodesic approximation in
AdS3, which reproduces the RT formula with using the geodesic length.

For a holographic BCFT, the two-point functions at leading order in the large
central charge limit $c\to\infty$ are dominated by two distinct channels:

$\expectationvalue{\sigma_{n}(t_{A_{1}},\theta_{A_{1}})\bar{\sigma}_{n}(t_{A_{2}},\theta_{A_{2}})}_{\mathrm{BCFT}}=\max\begin{cases}\expectationvalue{\sigma_{n}(t_{A_{1}},\theta_{A_{1}})\bar{\sigma}_{n}(t_{A_{2}},\theta_{A_{2}})}\\\
\expectationvalue{\sigma_{n}(t_{A_{1}},\theta_{A_{1}})}\expectationvalue{\bar{\sigma}_{n}(t_{A_{2}},\theta_{A_{2}})}\\\
\end{cases}\,.$ (3.18)

Here, the one-point function is non-vanishing due to the presence of a
boundary in the background. These two channels correspond to the connected and
disconnected Renyi entropies, analogous to the two types of RT surfaces in eq.
(3.5). For the standard AdS3/BCFT2 correspondence with an AdS brane, the
equivalence between eq. (3.18) and the holographic entanglement entropy
formula in eq. (3.5) can be demonstrated using the geodesic approximation.
However, our findings suggest that the result from the disconnected RT surface
anchored on a dS brane is problematic. In the following subsections, we will
demonstrate that the factorized two-point function is not given by the
disconnected extremal surface. Rather than employing the RT formula, we can
calculate the two-point function directly through the holographic
correspondence in AdS3. The connected channel remains consistent with that
derived from the connected RT surface. We will concentrate on the disconnected
phase, meticulously evaluating the one-point function of the twistor operators
in holographic field theory.

####  3.3.1 One-point functions in AdS/BCFT

Let us first consider Euclidean AdSd+1 bulk spacetime with a generic EOW
brane. For example, the Euclidean AdS metric $G_{MN}$ in $(d+1)$ dimensions
with the Poincarè coordinate reads

$ds^{2}=G_{MN}\;dx^{M}dx^{N}=\frac{L_{\textrm{\tiny
AdS}}^{2}}{z^{2}}\left(dz^{2}+\delta_{ij}dx^{i}dx^{j}\right)\,,$ (3.19)

where the BCFT is living on the conformal boundary with the Dirichlet boundary
condition and the EOW brane, satisfying the Neumann boundary condition, is
assumed to be given by the hypersurface $z=F(x_{b})$ with $x_{b}$ as the
induced metric on the EOW brane. The expressions for the other coordinates for
AdS${d+1}$ can be obtained by performing the coordinate transformations. Even
though there is an assumed EOW brane in the bulk space, the bulk-to-bulk
propagator $G{\textrm{\tiny BB}}^{\Delta}(x,x^{\prime})$ for the bulk scalar
field still satisfies the same equation of motion as that for the AdS space,
namely

$\left(\square_{G}+m^{2}\right)G_{\textrm{\tiny
BB}}^{\Delta}(x;x^{\prime})=\frac{\delta^{(d+1)}(x-x^{\prime})}{\sqrt{G}}\,,$
(3.20)

where the mass of the bulk field is related to the conformal dimension of the
dual operator on the boundary via the standard AdS/CFT dictionary
$m^{2}=\Delta(\Delta-d)$. For $(d+1)$-dimensional AdS space, the solution is
known as Witten:1998qj ; Freedman:1998tz ; Ammon:2015wua

$G_{\textrm{\tiny BB}}^{\Delta}(x;x^{\prime})=G_{\textrm{\tiny
BB}}^{\Delta}(\xi)=\frac{C_{\Delta}}{2^{\Delta}(2\Delta-d)}\xi^{\Delta}\cdot{}_{2}F_{1}\left(\frac{\Delta}{2},\frac{\Delta+1}{2};\Delta-\frac{d}{2}+1;\xi^{2}\right)\,,$
(3.21)

with the normalization constant given by

$C_{\Delta}=\frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma\left(\Delta-\frac{d}{2}\right)}\,.$
(3.22)

Here we have introduced the so-called chordal distance $\xi$ that is
associated with the (dimensionless) geodesic distance $D(x,x^{\prime})$ as the
following

$D(x;x^{\prime})=\,\ln\left(\frac{1+\sqrt{1-\xi^{2}}}{\xi}\right)\,,\qquad\xi=\frac{1}{\cosh\left(D(x;x^{\prime})\right)}.$
(3.23)

For the case with one point on the conformal boundary, e.g., $z=0$ in the
Poincaré coordinates, it is more useful to define the bulk-to-boundary
propagator $K_{\textrm{\tiny Bb}}^{\Delta}(x,x^{\prime})$ by taking the
boundary limit of the bulk-to-bulk propagator. For example, in the Poincare
coordinate, we have

$K_{\textrm{\tiny
Bb}}^{\Delta}(\vec{x};z^{\prime},\vec{x}^{\prime})=\lim_{z\rightarrow
0}\frac{2\Delta-d}{(z)^{\Delta}}G_{\textrm{\tiny
BB}}^{\Delta}(z,\vec{x};z^{\prime},\vec{x}^{\prime})\,,$ (3.24)

where the normalization factor is chosen to be the same as most literature.
The bulk-to-boundary propagator in other coordinates is obtained from the
conformal transformations.

The biggest advantage we take from the holographic bulk spacetime is that the
one-point function of BCFT on the conformal boundary can be derived using the
bulk-to-boundary propagator as follows Suzuki:2022xwv ; Izumi:2022opi ;
Kastikainen:2021ybu :

$\expectationvalue{\mathcal{O}(x)}_{\textrm{\tiny
BCFT}}=\alpha\cdot\,\int_{\textrm{\tiny
EOW}}d^{d}x_{b}\sqrt{g}K_{\textrm{\tiny Bb}}^{\Delta}(x;x_{b})\,.$ (3.25)

where $\alpha$ represents the normalization factor which is ignored for
simplicity, and the coordinates $x$ and $x_{b}$ denote the two points on the
Dirichlet boundary and the EOW brane, respectively. Due to this holographic
dictionary, we can then evaluate the one-point function of the twistor
operator and thus obtain the corresponding entanglement entropy in the
disconnected phase by applying the replica trick (3.17).

The integral along the EOW brane indicates that the gravitational region on
the brane is correlated with the boundary. It seems that this contradicts the
island story because the correlation is smeared over the whole EOW brane.
However, it is interesting to note that this integral along the brane could be
dominated by a particular saddle point, which can reproduce the island
formula. This saddle point approximation is essentially the known geodesic
approximation for the correlation functions in the AdS/CFT duality.

The first key point is that we only need correlation functions for the
operator with a large conformal dimension $\Delta\to\infty$ to evaluate the
entanglement entropy (3.18) using the twistor operators with
$\Delta_{n}\to\infty$. By taking $\Delta\to\infty$, we can approximate the
bulk-to-bulk propagator and the bulk-to-boundary propagator by the geodesic
distance Balasubramanian:1999zv , viz,

$\begin{split}G_{\textrm{\tiny
BB}}^{\Delta}(x;x_{b})&\approx\frac{1}{2\Delta-d}e^{-\Delta
D(x;x_{b})}e^{1-\frac{d}{2}}\quantity(\frac{\Delta}{2\pi}\frac{1+\sqrt{1-\xi^{2}}}{\sqrt{1-\xi^{2}}})^{\frac{d}{2}}\,,\\\
K_{\textrm{\tiny Bb}}^{\Delta}(\vec{x};z_{b},\vec{x}_{b})&=\lim_{z\rightarrow
0}e^{-\Delta\quantity(D(x;x_{b})+\log{z})}e^{1-\frac{d}{2}}\quantity(\frac{\Delta}{\pi})^{\frac{d}{2}}\,.\end{split}$
(3.26)

In this limit, the bulk-to-bulk and bulk-to-boundary propagators become
dominated by the geodesic distances, which simplifies the calculation of the
one-point function and, consequently, the entanglement entropy. For the
derivation of these expressions, see appendix B.

The one-point function of the primary operator on the conformal boundary of
BCFT is thus approximated by

$\expectationvalue{\mathcal{O}(x)}_{\textrm{\tiny
BCFT}}\approx\int_{\textrm{\tiny
EOW}}d^{d}x_{b}\sqrt{g}e^{-\Delta\bar{D}(x;x_{b})}\,,$ (3.27)

where $\bar{D}(x;x_{b})$ denotes the renormalized geodesic distance, excluding
the UV divergent part. To calculate the entanglement entropy, we apply this
large dimension limit to the twist operator $\sigma_{n}$ with conformal
dimension $\Delta_{n}=\frac{c}{12}\left(n-\frac{1}{n}\right)$. Considering the
holographic field theory with $c\to\infty$, the non-vanishing one-point
function is evaluated as

$\expectationvalue{\sigma_{n}(x_{A})}\approx\text{const.}\times\int_{\textrm{\tiny
EOW}}d^{2}x_{b}\sqrt{g}e^{-\Delta_{n}{D}(x_{A};x_{b})}\,,$ (3.28)

where we use the original geodesic length $D(x_{A},x_{b})$ to match the UV
divergence for the holographic entanglement entropy.

The second key approach is the saddle point approximation for evaluating the
integral along the brane. Assuming there exists a saddle point satisfying

$\partial_{x_{b}}D(x_{A};x_{b})\big{|}_{x_{b}=x_{b}^{\ast}}=0\,,$ (3.29)

along all directions on the EOW brane, the saddle point approximation999A
simple example of the saddle point approximation is the integral of the form
$I\equiv\int_{-\infty}^{\infty}e^{-f(x)}\,dx\approx
e^{-f\left(x_{0}\right)}\int_{-\infty}^{\infty}e^{-\frac{1}{2}\left(x-x_{0}\right)^{2}f^{\prime\prime}\left(x_{0}\right)}\,dx=e^{-f\left(x_{0}\right)}\sqrt{\frac{2\pi}{f^{\prime\prime}\left(x_{0}\right)}}\,.$
(3.30) This approximation only holds for locally minimal points where
$f^{\prime}\left(x_{0}\right)=0\,,\qquad
f^{\prime\prime}\left(x_{0}\right)>0\,.$ (3.31) applied to the integral
defined in eq. (3.28) yields 101010The volume measure of the brane, i.e.,
$\sqrt{g(x_{b})}$ would also influence the position of the saddle point.
However, its effect can be ignored after taking the limit
$\Delta_{n}\to\infty$.

$\expectationvalue{\sigma_{n}(x_{A})}\approx\sqrt{\frac{g(x_{b}^{*})}{\det(D^{\prime\prime}(x_{A};x_{b}^{\ast}))}}\;e^{-\Delta_{n}{D}(x_{A};x_{b}^{\ast})}\,.$
(3.32)

Finally, we find that this approximation of the one-point function reproduces
the holographic entanglement entropy for the disconnected extremal surface,
i.e.,

$\begin{split}S_{\mathcal{A}}&\approx\lim_{n\to
1}\frac{1}{1-n}\left(\lim_{\Delta_{n}\to\infty}\left(\log\expectationvalue{\sigma_{n}(x_{A_{1}})}+\log\expectationvalue{\sigma_{n}(x_{A_{2}})}\right)\right)\\\
&=\frac{c}{6}\,\left(D(x_{A_{1}};x_{b_{2}}^{\ast})+D(x_{A_{2}};x_{b_{2}}^{\ast})\right)\,,\\\
\end{split}$ (3.33)

where we take the leading contribution in the limit $\Delta_{n}\sim
c\to\infty$. The saddle point associated with the integral on the EOW brane
corresponds to the extremal geodesic orthogonal to the EOW brane. This saddle
point condition is equivalent to the extremization condition defined in eq.
(3.9). For a detailed illustration of this saddle point approximation in the
AdS/BCFT correspondence, see appendix B.

However, a caveat associated with the saddle point approximation for the
integral on the EOW brane is that it only applies to locally minimal points
satisfying

$\partial_{x_{b}}\partial_{x_{b}}{D}(x_{A};x_{b})\big{|}_{x_{b}=x_{b}^{\ast}}=D^{\prime\prime}(x_{A};x_{b})>0\,,$
(3.34)

or all directions along the EOW brane 111111More precisely, this condition
requires that the matrix
$\partial_{x_{b}^{\prime}}\partial_{x_{b}}{D}(x_{A},x_{b})\big{|}_{x_{b}=x_{b}^{\ast}}$
is positive definite.. Saddle points that do not satisfy this condition cannot
dominate the integral along the EOW brane. In our doubly holographic models
with a dS EOW brane, we will demonstrate that the saddle point approximation
fails because there are no locally minimal points on the dS EOW brane.

####  3.3.2 One-point function with the dS brane

In the following, we focus on computing the entanglement entropy with a dS
gravity by using the one-point function of the twistor operator, which avoids
the issues of the standard island formula and the disconnected RT surface. We
begin with the Euclidean AdS3/dS2 slicing coordinates:

$ds^{2}=L_{\textrm{\tiny
AdS}}^{2}\left(d\eta^{2}+\sinh^{2}{\eta}\left(dt_{\textrm{\tiny
E}}^{2}+\cos^{2}{t_{\textrm{\tiny E}}}d\theta^{2}\right)\right)\,,$ (3.35)

where the corresponding chordal distance is given by

$\xi=\frac{2\sech({\eta_{1}})\sech(\eta_{2})}{1-\tanh(\eta_{1})\tanh(\eta_{2})\left(\cos(t_{\textrm{\tiny
E}1})\cos(t_{\textrm{\tiny
E}2})\cos(\theta_{1}-\theta_{2})+\sin(t_{\textrm{\tiny
E}1})\sin(t_{\textrm{\tiny E}2})\right)}\,.$ (3.36)

We assume that the dS EOW brane with the Neumann boundary condition is located
at $\eta=\eta_{b}$, as depicted in figure 8. The induced metric on the
(Euclidean) EOW brane is given by the sphere metric:

$ds^{2}\big{|}_{\textrm{\tiny EOW}}=L_{\textrm{\tiny
AdS}}^{2}\sinh^{2}{\eta_{b}}\left(dt_{\textrm{\tiny
E}}^{2}+\cos^{2}{t_{\textrm{\tiny E}}}d\theta^{2}\right)\,.$ (3.37)

with the dS radius given by $L\equiv L_{\textrm{\tiny AdS}}\sinh{\eta_{b}}$.

Substituting the chordal distance to the bulk-to-boundary propagator
$K_{\textrm{\tiny Bb}}^{\Delta}$ defined in eq. (3.24) with $d=2$, one can
derive the non-vanishing one-point function (3.25) on the conformal boundary
at $\eta_{\infty}\to\infty$ by performing the integral along the Euclidean dS
EOW brane (i.e., a half sphere), viz,

$\expectationvalue{\mathcal{O}(t_{\textrm{\tiny E}},\theta)}_{\textrm{\tiny
dS}}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta_{b}\,\frac{\cos{t_{\textrm{\tiny
Eb}}}}{\quantity(\cosh{\eta_{b}}-\sinh{\eta_{b}}(\cos{t_{\textrm{\tiny
Eb}}}\cos{t_{\textrm{\tiny E}}}\cos(\theta-\theta_{b})+\sin{t_{\textrm{\tiny
E}b}}\sin{t_{\textrm{\tiny E}}}))^{\Delta}}\,,$ (3.38)

the normalization factor has been ignored and the edge of the EOW brane has
been chosen at $|\theta_{D}|=\frac{\pi}{2}$ for clarity. n particular, the
one-point function simplifies to

$\begin{split}\expectationvalue{\mathcal{O}(t_{\textrm{\tiny
E}},\theta)}_{\textrm{\tiny
dS}}\approx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta_{b}\,\frac{\cos
t_{\textrm{\tiny Eb}}}{\quantity(1-\cos t_{\textrm{\tiny E}}\cos
t_{\textrm{\tiny Eb}}\cos(\theta-\theta_{b})-\sin t_{\textrm{\tiny E}}\sin
t_{\textrm{\tiny Eb}})^{\Delta}}\ \,,\end{split}$ (3.39)

for the simpler model where the dS EOW brane is placed at
$\eta_{b}=\eta_{\infty}\sim\infty$. In the following, we focus on deriving the
leading contribution of the one-point function in eq. (3.60) in the limit
$\Delta\to\infty$ for the purpose of computing the entanglement entropy.

Saddle point is a local maximum: In the limit of large $\Delta$, we apply the
standard geodesic approximation and use the saddle point approximation to
evaluate the integral for the one-point function
$\expectationvalue{\sigma_{n}}_{\textrm{\tiny dS}}$ of the twistor operator.
The relevant integral is approximately given by

$\expectationvalue{\mathcal{O}(t_{\textrm{\tiny E}},\theta)}_{\textrm{\tiny
dS}}\approx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta_{b}\;\cos{(t_{\mathrm{Eb}})}\;e^{-\Delta
D(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny Eb}},\theta_{b})}\,,$ (3.40)

where $D(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny Eb}},\theta_{b})$
denotes the geodesic distance between the endpoints on the boundary and the
EOW brane, which is the same as in eq. (3.6) following the Wick rotation.
Ignoring the volume measure on the EOW brane121212 In fact, the integration
contour can be deformed to the complex plane, allowing the identification of
complex saddles. However, the complex saddles are located at
${t_{\textrm{\tiny E}}}=\frac{\pi}{2}+\pi n,\;n\in\mathbb{Z}$.. This results
in a vanishing prefactor in eq. (3.50). Additionally, we observe a complex
geodesic length, which lacks a straightforward physical interpretation. In
this paper, we concentrate on directly evaluating the integral on the EOW
brane., we can find only one saddle point for the integral, i.e.,

$x_{b}^{\ast}=\left(\theta_{b}=\theta-\pi\,,t_{\textrm{\tiny
Eb}}=-t_{\textrm{\tiny E}}\right)\,,$ (3.41)

whose Lorentzian continuation corresponds to the endpoint of the disconnected
RT surface derived in eq. (3.11). This saddle point indeed represents the
antipodal island. However, it is straightforward to see that this antipodal
saddle point $x_{b}^{\ast}$ is a local maximum for the geodesic distance
function because of

$\frac{\partial^{2}D\left(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny
Eb}},\theta_{b}\right)}{\partial
t_{\mathrm{Eb}}^{2}}\bigg{|}_{x_{b}=x_{b}^{\ast}}<0,\quad\frac{\partial^{2}D\left(t_{\textrm{\tiny
E}},\theta;t_{\textrm{\tiny
Eb}},\theta_{b}\right)}{\partial\theta_{b}^{2}}\bigg{|}_{x_{b}=x_{b}^{\ast}}<0\,.$
(3.42)

This indicates that the saddle point approximation using the antipodal saddle
point at $x_{b}^{\ast}$ cannot be applied to the integral for calculating the
one-point function of the twistor operator. In other words, the island formula
cannot be applied using the quantum extremal surface anchored on the dS
gravitational space. This nullity of the saddle point approximation highlights
the issues associated with the island formula and the disconnected extremal
surface in dS gravity.

Figure 9: The integrand $\sqrt{g(x_{b})}K_{\textrm{\tiny
Bb}}^{\Delta}(x;x_{b})$ for evaluating the one-point function
$\expectationvalue{\mathcal{O}(t_{\textrm{\tiny E}},\theta)}_{\textrm{\tiny
dS}}$ defined in eq. (3.60) is a function of the endpoint $(t_{\textrm{\tiny
Eb}},\theta_{b})$ on the EOW brane. We take the boundary point at
$\theta=\frac{3\pi}{4},t_{\textrm{\tiny E}}=\frac{\pi}{4}$ with choosing
$\eta_{b}=1$. For two bottom plots with various values of $\Delta$, one more
parameter is chosen as $t_{\textrm{\tiny Eb}}=\frac{\pi}{4}$,
$\theta_{b}=\frac{\pi}{2}$, respectively.

Dominant contribution from the edge: Instead of using the geodesic and saddle
point approximations, we can derive an analytic expression for the
$\theta_{b}$ integral in terms of Appell hypergeometric functions. However, a
key observation is that the integral along the dS EOW brane is always
dominated by contributions from the edge, i.e., $\theta_{b}=\frac{\pi}{2}$ for
$\theta\in\left(\frac{\pi}{2},\pi\right)$ and $\theta_{b}=-\frac{\pi}{2}$ for
$\theta\in\left(-\pi,-\frac{\pi}{2}\right)$. This is illustrated in figure 9,
where the integrand $\sqrt{g(x_{b})}K_{\textrm{\tiny Bb}}^{\Delta}(x;x_{b})$
is plotted as a function of the brane point. Moreover, it is important to note
that this edge contribution becomes increasingly dominant as the operator
dimension $\Delta$ grows. Consequently, in the limit $\Delta\to\infty$, the
spatial integral over $\theta_{b}$ can be approximated by its value at the
edge of the EOW brane. This can be expressed as:

$\begin{split}\lim_{\Delta\to\infty}\expectationvalue{\mathcal{O}(t_{\textrm{\tiny
E}},\theta)}_{\textrm{\tiny
dS}}&=\lim_{\Delta\to\infty}\left(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta_{b}\sqrt{g(t_{\textrm{\tiny
Eb}},\theta_{b})}K_{\textrm{\tiny Bb}}^{\Delta}(t_{\textrm{\tiny
E}},\theta;t_{\textrm{\tiny Eb}},\theta_{b})\right)\\\
&\approx\lim_{\Delta\to\infty}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\left(\sqrt{g(t_{\textrm{\tiny Eb}},\theta_{b})}K_{\textrm{\tiny
Bb}}^{\Delta}(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny
Eb}},\theta_{b})\right)\bigg{|}_{|\theta_{b}|=\frac{\pi}{2}}\\\
&=\lim_{\Delta\to\infty}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\,\frac{\cos{t_{\textrm{\tiny
Eb}}}}{\left(\cosh{\eta_{b}}-\sinh{\eta_{b}}(\cos{t_{\textrm{\tiny
Eb}}}\cos{t_{\textrm{\tiny E}}}|\sin\theta|+\sin{t_{\textrm{\tiny
E}b}}\sin{t_{\textrm{\tiny E}}})\right)^{\Delta}}\,,\end{split}$ (3.43)

where the sign depends on the range of $\theta$. By focusing on the
contributions from the edge, we simplify the calculation and capture the
dominant behavior of the integral in the large $\Delta$ limit. We have also
conducted a more detailed examination of the errors resulting from the
utilization of this non-saddle point approximation. Our findings show that the
sub-leading terms for computing the entanglement entropy are negligible in
comparison to the leading term given by the edge. Thus, we achieve a reliable
and precise evaluation of the one-point function with $\Delta\to\infty$,
despite not relying on the traditional saddle point approximation. This edge-
dominant method is particularly useful in the context of dS gravity, where the
standard saddle point approach encounters significant challenges.

Once the approximation for the spatial integral has been taken, it becomes
more straightforward to track the integral along the Euclidean time direction
on the EOW brane, given that the integrand behaves in a manner akin to a
Gaussian function, as shown in figure 9. In other words, we can then apply the
saddle point approximation to the $t_{\textrm{\tiny Eb}}$-integral. The time
derivative of the integrand in eq. (3.43) is derived as

$\frac{-\Delta\sinh\eta_{b}\left(\sin t_{\textrm{\tiny E}}\cos
t_{\textrm{\tiny Eb}}-\sin t_{\textrm{\tiny Eb}}\cos t_{\textrm{\tiny
E}}|\sin\theta|\right)}{\left(\cosh{\eta_{b}}-\sinh{\eta_{b}}(\cos{t_{\textrm{\tiny
Eb}}}\cos{t_{\textrm{\tiny E}}}|\sin\theta|+\sin{t_{\textrm{\tiny
E}b}}\sin{t_{\textrm{\tiny E}}})\right)}+\tan t_{\textrm{\tiny
Eb}}+\mathcal{O}\left(\frac{1}{\Delta}\right)\,,$ (3.44)

where we have performed a series expansion with for large $\Delta$ and keep
the first two leading orders. At leading order, one can get one saddle point
located at

$\tan t_{\textrm{\tiny Eb}}^{\ast}=\frac{\tan t_{\textrm{\tiny
E}}}{|\sin\theta|}\,\,,$ (3.45)

which corresponds the peak appearing in the $t_{\textrm{\tiny Eb}}$-integral.
One may worry that the volume measure of the EOW brane may influence this
saddle point approximation due to the fact that Euclidean brane is a compact
sphere. It is indeed true that there are two extra saddle points at

$t_{\textrm{\tiny
Eb}}\approx\frac{\pi}{2}+\frac{1}{\Delta}\frac{\cosh\eta_{b}-\sin
t_{\textrm{\tiny Eb}}}{|\sin\theta|\cos t_{\textrm{\tiny
Eb}}}+\mathcal{O}\left(\frac{1}{\Delta^{2}}\right)\quad\text{and}\quad
t_{\textrm{\tiny
Eb}}\approx-\frac{\pi}{2}+\frac{1}{\Delta}\frac{\cosh\eta_{b}+\sin
t_{\textrm{\tiny Eb}}}{|\sin\theta|\cos t_{\textrm{\tiny
Eb}}}+\mathcal{O}\left(\frac{1}{\Delta^{2}}\right)\,,$ (3.46)

which are close to the edge of the time direction. However, these saddle
points can only appear and dominate the integral in the regime with $\cos
t_{\textrm{\tiny E}}<0$ which is not included by the Hartle-Hawking state with
a compact Euclidean time $t_{\textrm{\tiny
E}}\in(-\frac{\pi}{2},\frac{\pi}{2})$.

Figure 10: Numerical integrals of the one-point function
$\frac{-\log\expectationvalue{\mathcal{O}(t_{\textrm{\tiny
E}},\theta)}_{\textrm{\tiny dS}}}{\Delta}$ with different values of the
dimension $\Delta$. As $\Delta$ increases, the one-point function approaches
the analytical expression derived in eq. (3.47) for $\Delta\to\infty$. For
these two numerical plots, we choose $\eta_{b}=2$ and set $t_{\textrm{\tiny
E}}=0$ for the left plot and $\theta=\frac{3\pi}{4}$ for the right plot.

As a summary, we conclude that the $t_{\textrm{\tiny Eb}}$-integral for the
one-point function (3.43) at the large dimension limit is dominated by the
saddle point located at $t_{\textrm{\tiny Eb}}=t_{\textrm{\tiny Eb}}^{\ast}$.
We are interested in computing the entanglement entropy by using the one-point
function of the twistor operator as defined in eq. (3.17). More explicitly, we
can get

$\begin{split}\lim_{\Delta\to\infty}\left(\frac{-\log\expectationvalue{\mathcal{O}(t_{\textrm{\tiny
E}},\theta)}_{\textrm{\tiny
dS}}}{\Delta}\right)&\approx\log\left(\cosh{\eta_{b}}-\sinh{\eta_{b}}(\cos{t^{\ast}_{\textrm{\tiny
Eb}}}\cos{t_{\textrm{\tiny E}}}|\sin\theta|+\sin{t^{\ast}_{\textrm{\tiny
E}b}}\sin{t_{\textrm{\tiny E}}})\right)\\\
&=\log\left(\cosh\eta_{b}-\sinh{\eta_{b}}\sqrt{\cos^{2}t_{\textrm{\tiny
E}}\sin^{2}\theta+\sin^{2}t_{\textrm{\tiny E}}}\right)\,,\end{split}$ (3.47)

where the sub-leading terms all vanish in the limit $\Delta\to\infty$. By
comparing with the numerical integrals, we can find that the one-point
function approaches the analytical expression (3.47) by taking large dimension
limit, as shown in figure 10.

Equipped with the result of the one-point function
$\mathcal{O}\left(t_{\textrm{\tiny E}},\theta\right)$ in the large dimension
limit, we can finally apply the replica trick (3.17) to calculate the
entanglement entropy of a boundary interval. Taking the symmetric interval
$\mathcal{A}$ (2.9) on the Euclidean boundary, i.e.,

$\mathcal{A}:\qquad\\{t_{\textrm{\tiny E}}=t_{\textrm{\tiny
E}A}\,,\quad\theta\in[-\pi,-\theta_{A}]\cup[\theta_{A},\pi]\\}\,,$ (3.48)

the entanglement entropy $S_{\mathcal{A}}$ derived from the replica trick and
double holography is given by the following minimization

$S_{\mathcal{A}}=\min\begin{cases}S_{\mathcal{A}}^{\rm
con}=\frac{c}{3}\log\left(\frac{2\cos t_{\textrm{\tiny
E}A}\cdot\sin\theta_{A}}{\epsilon}\right)\\\ \,\\\
\tilde{S}_{\mathcal{A}}^{\rm
dis}=\frac{c}{3}\log\left(\frac{2}{\epsilon}\left(\cosh\eta_{b}-\sinh{\eta_{b}}\sqrt{\cos^{2}t_{\textrm{\tiny
E}A}\sin^{2}\theta_{A}+\sin^{2}t_{\textrm{\tiny E}A}}\right)\right)\\\
\end{cases}\,,$ (3.49)

where we have restored the universal divergent part for the disconnected phase
$S_{\mathcal{A}}^{\rm dis}$. When the subsystem $\mathcal{A}$ is large (i.e.,
$\theta_{A}$ is close to $\frac{\pi}{2}$), the disconnected phase is always
favored. At late times $t_{\textrm{\tiny E}A}\sim\frac{\pi}{2}$, the connected
phase with $S_{\mathcal{A}}^{\rm con}$ would be always dominant since it
approaches negative infinity. These phase transition are illustrated by the
example shown in figure 12.

Figure 11: The entanglement entropy $S_{\mathcal{A}}$ derived in eq. (3.56) is
given by the minimum value between the connected phase
$S_{\mathcal{A}}^{\mathrm{con}}$ and the disconnected phase
$\tilde{S}_{\mathcal{A}}^{\mathrm{dis}}$. In both plots we subtract the UV
divergence $\frac{c}{3}\log{\frac{2}{\epsilon}}$ from the entanglement entropy
and take $\eta_{b}=2$. For the left/right plot we set $t_{\textrm{\tiny
E}A}=0$ and $\theta_{A}=2$, respectively.

###  3.4 Holographic dual and non-extremal island on the dS brane

$\Gamma_{\mathcal{A}}^{\mathrm{con}}$$\mathcal{A}$EWSmall $\mathcal{A}$

$\tilde{\Gamma}_{\mathcal{A}}^{\mathrm{dis}}$$\tilde{\Gamma}_{\mathcal{A}}^{\mathrm{dis}}$$\mathcal{A}$Non-
extremal IslandEWLarge $\mathcal{A}$

Figure 12: Left: The non-island phase for a small interval $\mathcal{A}$.
Right: The island phase for a large interval $\mathcal{A}$ coupled to the dS
gravity. Note that the non-extremal island, indicated by the blue curve,
includes the entire dS gravity region.

In the preceding subsection, we computed the entanglement entropy
$S_{\mathcal{A}}$ utilizing the holographic correlation functions of the
twistor operators. Although we have previously demonstrated that neither the
standard holographic entanglement entropy formula, which involves extremal
surfaces, nor the conventional island formula is applicable to a subsystem
coupled with dS gravitational spacetime, our results obtained from the
holographic correlation functions suggest that a novel holographic formula for
evaluating entanglement entropy may be identified.

Despite the limitations of the saddle point approximation, we continue to
derive the holographic entanglement entropy at leading order with
$\frac{1}{G_{\textrm{\tiny N}}}\sim c\to\infty$. Consequently, both the one-
point function and two-point function of the twistor operator at large
dimensions are consistently described by the geodesic approximation, as
discussed around eq. (3.27). For instance, the one-point function
$\expectationvalue{\sigma_{n}}_{\textrm{\tiny dS}}$ for $\Delta_{n}\to\infty$
can be expressed as

$\lim_{\Delta_{n}\to\infty}\expectationvalue{\sigma_{n}(t_{\textrm{\tiny
E}},\theta)}_{\textrm{\tiny
dS}}\approx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta_{b}\;\cos{(t_{\mathrm{Eb}})}\;e^{-\Delta_{n}D(t_{\textrm{\tiny
E}},\theta;t_{\textrm{\tiny Eb}},\theta_{b})}\,,$ (3.50)

where the geodesic distance $D(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny
Eb}},\theta_{b})$ from a boundary point to the Euclidean EOW brane is given by

$D(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny
Eb}},\theta_{b})=\eta_{\infty}+\log{\quantity(\cosh\eta_{b}-\sinh{\eta_{b}}\quantity(\cos{{t_{\textrm{\tiny
E}}}}\cos{{t_{\textrm{\tiny
E}}}_{b}}\cos(\theta-\theta_{b})+\sin{{t_{\textrm{\tiny
E}}}}\sin{{t_{\textrm{\tiny E}}}_{b}}))}\,.$ (3.51)

By applying the insights gained from evaluating the two integrals associated
with the one-point function, we find that the dominant contribution is
equivalent to the length of the geodesic connecting the boundary point to the
brane point at the edge with $|\theta_{b}|=\frac{\pi}{2}$, i.e.,

$\tilde{S}_{\mathcal{A}}^{\rm dis}=\frac{L_{\textrm{\tiny
AdS}}}{4G_{\textrm{\tiny N}}}\left(D(t_{\textrm{\tiny
E}A},\theta_{A};t_{\textrm{\tiny
Eb}},\theta_{b})\bigg{|}_{\theta_{b}=\frac{\pi}{2},t_{\textrm{\tiny
E}b}=t^{\ast}_{\textrm{\tiny E}b}}+D(t_{\textrm{\tiny
E}A},-\theta_{A};t_{\textrm{\tiny
Eb}},\theta_{b})\bigg{|}_{\theta_{b}=-\frac{\pi}{2},t_{\textrm{\tiny
E}b}=t^{\ast}_{\textrm{\tiny E}b}}\right)\,,$ (3.52)

where the time slice $t_{\textrm{\tiny Eb}}^{\ast}$ on the brane is determined
by

$\tan t_{\textrm{\tiny Eb}}^{\ast}=\frac{\tan t_{\textrm{\tiny
E}A}}{|\sin\theta_{A}|}\,.$ (3.53)

It is important to note that this type of disconnected geodesic, denoted as
$\tilde{\Gamma}^{\rm dis}_{\mathcal{A}}$, differs from the extremal geodesic
${\Gamma}^{\rm dis}_{\mathcal{A}}$ anchored on the EOW brane. The disconnected
geodesic is non-extremal with respect to the spatial direction but is locally
minimal with respect to variations along the Euclidean timelike direction on
the EOW brane. It is straightforward to verify that its second time derivative
is always positive:

$\frac{\partial^{2}D(t_{\textrm{\tiny E}},\theta;t_{\textrm{\tiny
Eb}},\theta_{b})}{\partial t_{\textrm{\tiny
E}b}^{2}}\bigg{|}_{\theta_{b}=\frac{\pi}{2},t_{\textrm{\tiny
E}b}=t^{\ast}_{\textrm{\tiny
E}b}}=\frac{1}{\frac{\coth{\eta_{b}}}{\sqrt{\sin^{2}{t_{\textrm{\tiny
E}}}+\sin^{2}{\theta}\cos^{2}{t_{\textrm{\tiny
E}}}}}-1}>\frac{1}{\coth{\eta_{b}}-1}>0\,.$ (3.54)

In summary, as illustrated in figure 12, the key result from the calculation
of the one-point function$\expectationvalue{\sigma_{n}(t_{\textrm{\tiny
E}},\theta)}_{\textrm{\tiny dS}}$ is the derivation of a new holographic
entanglement formula for an interval connected to the dS brane. Specifically,
we have

$S_{\mathcal{A}}=\min\quantity{S_{\mathcal{A}}^{\mathrm{con}},\tilde{S}_{\mathcal{A}}^{\mathrm{dis}}}=\min\left\\{\frac{\text{Area}(\Gamma_{\mathcal{A}}^{\rm{con}})}{4G_{\textrm{\tiny
N}}},\frac{\text{Area}(\tilde{\Gamma}_{\mathcal{A}}^{\rm{dis}})}{4G_{\textrm{\tiny
N}}}\right\\}\,,$ (3.55)

where the disconnected surface $\tilde{\Gamma}_{\mathcal{A}}^{\rm{dis}}$,
which connects the endpoints of the boundary interval and the edge of the dS
EOW brane, is extremal only with respect to the time direction on the dS
brane.

Finally, we turn our attention to the island formula for a non-gravitational
bath coupled to de Sitter gravity. It remains intriguing to interpret the
disconnected phase as the island phase from the perspective of the brane.
However, in this case, the boundary of the island is no longer determined by
the quantum extremal surface. Our findings thus indicate the presence of a
non-extremal island whose boundary is fixed at the edge of the dS
gravitational region. In the connected phase, or non-island phase, where the
size $\mathcal{A}$ is small, the entanglement wedge does not extend to the EOW
brane. Consequently, an observer situated within $\mathcal{A}$ cannot probe
the dS gravitational region. As the size of the bath interval$\mathcal{A}$
increases, a phase transition occurs from the non-island phase to the island
phase. Thus, in the island phase, the entanglement wedge of the non-
gravitational interval will always encompass the entire dS gravitational
region. This marks a clear distinction between dS gravity and AdS gravity,
where a portion of the AdS gravitational region can still be entangled with a
non-gravitational bath.

###  3.5 Lorentzian dS brane

Figure 13: The Lorentzian time evolution of the connected and disconnected
holographic entanglement entropy. We have subtracted the UV divergence
$\frac{c}{3}\log{\frac{2}{\epsilon}}$. In the left and right panels, we set
the subsystem to be $\mathcal{A}=[\theta=0.6\pi,-0.6\pi]$ and
$\mathcal{A}=[\theta=0.7\pi,-0.7\pi]$, respectively, with $\eta_{b}=2$. After
the critical time $t_{A}=t_{\mathrm{cr}}$, denoted by the dashed line, we plot
the real part of the disconnected holographic entanglement entropy.

The analysis in the previous section focused on the Euclidean holographic
correlation function with a Euclidean EOW brane, from which we derived the
holographic entanglement entropy $S_{\mathcal{A}}$. Finally, we turn to the
Lorentzian time evolution of holographic entanglement entropy. By taking the
Euclidean result $S_{\mathcal{A}}$ derived in eq. (3.56) and performing the
Wick rotation $t_{\textrm{\tiny E}A}=it_{A}$, we obtain

$S_{\mathcal{A}}=\min\begin{cases}S_{\mathcal{A}}^{\rm
con}=\frac{c}{3}\log\left(\frac{2\cosh
t_{A}\cdot\sin\theta_{A}}{\epsilon}\right)\\\ \,\\\
\tilde{S}_{\mathcal{A}}^{\rm
dis}=\frac{c}{3}\log\left(\frac{2}{\epsilon}\left(\cosh\eta_{b}-\sinh{\eta_{b}}\sqrt{\cosh^{2}t_{A}\sin^{2}\theta_{A}-\sinh^{2}t_{A}}\right)\right)\\\
\end{cases}\,.$ (3.56)

However, it is obvious that this disconnected phase
$\tilde{S}_{\mathcal{A}}^{\rm dis}$ becomes complex valued after the critical
time $t_{\mathrm{cr}}$ given by

$\tanh{t_{\mathrm{cr}}}=|\sin{\theta_{A}}|.$ (3.57)

To understand the origin of the critical time, we note that the saddle point
on the time direction along the EOW brane (3.53), i.e., the endpoint of the
geodesic $\tilde{\Gamma}_{\mathcal{A}}^{\rm dis}$ reads

$\tanh t_{b}^{\ast}=\frac{\tanh t_{A}}{|\sin\theta_{A}|}\,,$ (3.58)

after the Wick rotation. We found that the endpoints of the disconnected
geodesics $\tilde{\Gamma}_{\mathcal{A}}^{\rm dis}$ on the dS EOW branes reach
future infinity $t_{b}\to\infty$ at the critical time $t_{A}=t_{\mathrm{cr}}$.
The fact that the geodesic length becomes complex-valued for
$t_{A}>t_{\mathrm{cr}}$ indicates that the geodesic extends into the
complexified coordinate direction. In this situation, as argued in the
holographic dual of a half de Sitter space Kawamoto:2023nki , we can regard
the real part of $\tilde{S}_{\mathcal{A}}^{\mathrm{dis}}$ as the entanglement
entropy, assuming the Schwinger-Keldysh formulation of the time-dependent
density matrix. With an appropriate future projection to a final state, we may
also interpret the complex entropy $\tilde{S}_{\mathcal{A}}^{\mathrm{dis}}$ as
holographic pseudo entropy Nakata:2020luh  rather than holographic
entanglement entropy, which is analogous to time-like entanglement entropy
Doi:2022iyj ; Doi:2023zaf . In the former interpretation, we plotted the
Lorentzian time evolution of both the connected and disconnected contributions
in figure 13. For a very large subsystem $\mathcal{A}$, where its endpoints
are close to the edges at $\theta=\pm\frac{\pi}{2}$, the disconnected phase is
always dominant. It exhibits a monotonic growth until the critical time. After
the critical time, it grows linearly as $\sim\frac{c}{6}t_{A}$, as shown in
figure 13. As the size of $\mathcal{A}$ decreases, the connected phase can
compete with the disconnected phase, leading to phase transitions as depicted
in the right panel of figure 13. It is noteworthy that, regardless of the
dominant phase, there is a linear growth in $\mathcal{A}$ at late times, i.e.,

$S_{\mathcal{A}}=\min\quantity{S_{\mathcal{A}}^{\rm
con},\text{Re}\left(\tilde{S}_{\mathcal{A}}^{\rm
dis}\right)}\,\sim\,\frac{c}{3}t_{A}\,,\quad\text{for}\quad t_{A}\gg 1\,.$
(3.59)

This can be attributed to the inflation of dS space in global time, which
affects the entanglement entropy through the Weyl factor.

Given the success of the holographic one-point function in computing the
holographic entanglement entropy within the Euclidean signature, there is
reason to believe that the Lorentzian one-point function of the twistor
operator may prove to be an effective means of determining the entanglement
entropy at late times. To illustrate, the holographic one-point function
$\expectationvalue{\mathcal{O}(t,\theta)}_{\textrm{\tiny dS}}$ with the
Lorentzian signature can be expressed as follows

$\expectationvalue{\mathcal{O}(t,\theta)}_{\textrm{\tiny
dS}}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dt_{\textrm{\tiny
Eb}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta_{b}\,\frac{\cos{t_{\textrm{\tiny
Eb}}}}{\quantity(\cosh{\eta_{b}}-\sinh{\eta_{b}}(\cos{t_{\textrm{\tiny
Eb}}}\cosh t\cos(\theta-\theta_{b})+i\sin{t_{\textrm{\tiny E}b}}\sinh
t))^{\Delta}}\,.$ (3.60)

which is always real by definition131313The imaginary part is precisely
canceled due to the fact that the $t_{\textrm{\tiny E}b}$-integral is
symmetric between $\pm t_{\textrm{\tiny E}b}$. However, the above integral is
not well-defined due to the pole at $t=0$ slice. For example, we can find
explicit expression for some particular $\Delta$ while there is a branch cut
Nevertheless, the above integral is not well-defined due to the presence of a
pole at the $t=0$ slice. For instance, the explicit expression for some
specific value of $\Delta$ can be derived but with the presence of a branch
cut at $\cosh t=\frac{\coth\eta_{b}}{|\sin\theta|}$. It is anticipated that
the complex deformation of the integral contour may circumvent this issue and
yield the same result as the Wick rotation. Nevertheless, we discovered that
the result of the integral for
$\expectationvalue{\mathcal{O}(t,\theta)}_{\textrm{\tiny dS}}$ is highly
sensitive to the choice of the complex contour, and thus, We leave this issue
for future investigation.

##  4 Non-extremal island in dS or AdS braneworld

The island formula in the AdS setup where the gravitational theory is coupled
to the non-gravitational bath is captured by the quantum extremal surface,
which extremizes the generalized entropy denoted by $S_{\rm gen}$. In this
setup, the generalized entropy is maximized when the position of the island is
varied along the spatial direction and minimized when the position is varied
along the temporal direction. However, as previously demonstrated, the quantum
extremal surface in dS gravity coupled to the bath is instead minimized along
the time-like direction, but maximized along the spatial direction. This
compelling evidence suggests that this type of quantum extremal surface in dS
gravity is not a physically relevant quantity for calculating the correct
entanglement entropy. One possible explanation for the discrepancy between
quantum extremal surfaces in AdS gravity and dS gravity is the sign of the
Ricci curvature of the gravitational spacetime where the island resides.
However, this section will demonstrate that the stability of the extremal
surfaces is contingent upon the extrinsic curvature of the gravitational
brane-world from the perspective of the double holography. In a more generic
context, the stability of the quantum extremal surface is determined by the
second variation of the generalized entropy along the gravitational region. By
using the doubly holographic holographic setup, we can derive the generalized
entropy in the brane perspective as the area of the extremal surface anchored
on the two-dimensional (AdS or dS) brane-world living in the AdS3 bulk
spacetime. The double holography allows us to ascertain whether a quantum
extremal surface is a maximin surface by examining the variation of the
geodesic length along the EOW brane.

In section 4.1, we will examine the first and second variation of geodesic
length in an arbitrary Euclidean spacetime. By employing Synge’s formula for
the second variation of geodesics, we can derive a necessary and sufficient
condition for the stability of the extremal surface. The stable extremal
surface is referred to as the maximin surface in the Lorentzian signature and
minimal surface in the Euclidean signature. In our doubly holographic setup,
we will demonstrate that the sufficient condition is related to the extrinsic
curvature of the EOW brane. In section 4.2, we will examine explicit examples
in the AdS3/BCFT2 setup with time-symmetric profiles of EOW branes. It can be
demonstrated that both the dS brane and the AdS brane are susceptible to
unstable extremal surfaces.

###  4.1 Geodesic variation and stability condition

Figure 14: The geodesic variations of $C_{0}$ form a one-parameter family of
neighboring lines, $C_{z}$, parametrized by $z$. In particular, we are
interested in the variation along a brane, which is obtained by anchoring all
perturbed geodesics to the brane.

As motivated before, we first focus on exploring the variations of geodesic
length in a generic Euclidean spacetime in this subsection. The background
results in this subsection can also found in textbooks about Riemannian
geometry, e.g., frankel2011geometry ; chow2023hamilton ; Hawking:1973uf . Even
we do not show the results explicitly, similar results can be also derived for
the variation of area of codimension$-n$ surfaces in higher dimensional space.
We believe that it can be applied to explore the stability of generalized
entropy for a higher-dimensional system. We will leave this for future
exploration and focus on the geodesic variations for simplicity.

Supposing $C_{0}(s)$ is a curve in the Riemannian manifold with a given metric
$g(x^{\mu})$, we have parametrized the curve by a parameter $s$. To explore
the variation from the original curve $C_{0}(s)$, we can define a one-
parameter family of neighboring lines $C_{z}(s)$ parameterized by the
variation parameter $z$, see figure 14 for the illustration. The arc length
$L(C_{z})$ for a single curve is defined by

$L(C_{z})=\int_{0}^{1}\sqrt{g\left(\frac{\partial\mathbf{x}}{\partial
s},\frac{\partial\mathbf{x}}{\partial s}\right)}ds\,,$ (4.1)

where $g(\mathbf{v},\mathbf{v}^{\prime})$ denotes the inner product of two
vectors associated with the Riemannian metric $g$ and we have chosen the
geodesic parameter $s$ as the affine parameter. By defining the tangent vector
and variation vector as

$\mathbf{T}=\frac{\partial\mathbf{x}}{\partial
s},\qquad\mathbf{V}=\frac{\partial\mathbf{x}}{\partial z}\,,$ (4.2)

respectively, we can derive the first variation of the arc length is known as

$\frac{d}{dz}L(C_{z})\bigg{|}_{z=0}=\frac{1}{L_{0}}\left(g\left(\mathbf{t},\mathbf{v}\right)\bigg{|}_{0}^{1}-\int_{0}^{1}g\left(\nabla_{\mathbf{t}}\mathbf{t},\mathbf{v}\right)ds\right)\,,$
(4.3)

where the geodesic length reduces to $L_{0}=L(C_{0})=||\mathbf{t}||$,
$\mathbf{t}=\mathbf{T}|_{z=0}$ is the tangent vector along the curve $C_{0}$,
and $\mathbf{v}=\mathbf{V}|_{z=0}$ is the variation vector for the original
curve.

Since we are interested in the extremal surface, i.e., geodesics , the tangent
vector of the original curve should satisfy the geodesic equation, i.e.,
$\nabla_{\mathbf{t}}\mathbf{t}=0$. As a result, the first variation formula
for geodesics reduces to

$\frac{d}{dz}L(C_{z})\bigg{|}_{z=0}=\frac{1}{L_{0}}g\left(\mathbf{t},\mathbf{v}\right)\bigg{|}_{0}^{1}\,.$
(4.4)

where only the boundary terms at the endpoint contribute. Furthermore, the
second variation of the arc length of the geodesic is given by Synge’s
formula, namely141414Our definition of Riemannian tensor is given by
$R(\mathbf{X},\mathbf{Y})\mathbf{Z}=\nabla_{\mathbf{X}}\nabla_{\mathbf{Y}}\mathbf{Z}-\nabla_{\mathbf{X}}\nabla_{\mathbf{Y}}\mathbf{Z}-\nabla_{\left[\mathbf{X},\mathbf{Y}\right]}\mathbf{Z}\,.$
(4.5) It is also useful to define the component of any vector $\mathbf{X}$
perpendicular to tangent vector $\mathbf{t}$ as
$\mathbf{X}^{\bot}=\mathbf{X}-\frac{g\left(\mathbf{X},\mathbf{t}\right)}{||\mathbf{t}||^{2}}\mathbf{t}\,.$
(4.6)

$\frac{\partial^{2}L(C_{z})}{\partial z\partial
z}\bigg{|}_{z=0}=\frac{1}{L_{0}}g\left(\mathbf{t},\nabla_{\mathbf{v}}\mathbf{v}\right)\bigg{|}^{1}_{0}+\frac{1}{L_{0}}\int^{1}_{0}\left[||(\nabla_{\mathbf{t}}\mathbf{v})\wedge\mathbf{t}||^{2}-g\left(R(\mathbf{v},\mathbf{t})\mathbf{t},\mathbf{v}\right)\right]ds,$
(4.7)

where
$||(\nabla_{\mathbf{t}}\mathbf{v})\wedge\mathbf{t}||^{2}=g\left((\nabla_{\mathbf{t}}\mathbf{v})^{\bot},(\nabla_{\mathbf{t}}\mathbf{v})^{\bot}\right)$
is the area of the parallelogram spanned by the vectors
$\mathbf{t},\nabla_{\mathbf{t}}\mathbf{v}$ 151515For the second order
variation of geodesic, one can also introduce another independent parameter
$w$ with new variation vector $\mathbf{w}$ and derive the variation of arc
length in terms of  $\frac{\partial^{2}L(C_{z})}{\partial z\partial
w}\bigg{|}_{z=w=0}=\frac{1}{L_{0}}g\left(\mathbf{t},\nabla_{\mathbf{w}}\mathbf{v}\right)\bigg{|}^{1}_{0}+\frac{1}{L_{0}}\int^{1}_{0}\left[g\left((\nabla_{\mathbf{t}}\mathbf{v})^{\bot},(\nabla_{\mathbf{t}}\mathbf{w})^{\bot}\right)+g\left(R(\mathbf{w},\mathbf{t})\mathbf{v},\mathbf{t}\right)\right]ds\,.$
(4.8) In a more general sense, these variation formulas can be generalized to
the case with an arbitrary functional, leading to similar geometric
conclusions. For more details, see e.g., Bernamonti:2019zyy ;
Bernamonti:2020bcf ; chow2023hamilton  . It is obvious that the variation
proportional to the tangent vector is trivial since it does not change the
curve. In the following, we only need to consider the orthogonal variation
with satisfying $g(\mathbf{t},\mathbf{v})=0$ as well as
$g\left(\nabla_{\mathbf{t}}\mathbf{v},\mathbf{t}\right)=0$. Consequently, the
orthogonal variation for a geodesic reduces to

$L^{\prime\prime}(C_{0})=\frac{1}{L_{0}}g\left(\mathbf{t},\nabla_{\mathbf{v}}\mathbf{v}\right)\bigg{|}^{1}_{0}+\frac{1}{L_{0}}\int^{1}_{0}\left[||(\nabla_{\mathbf{t}}\mathbf{v})||^{2}-g\left(R(\mathbf{v},\mathbf{t})\mathbf{t},\mathbf{v}\right)\right]ds\,.$
(4.9)

By using this second variation formula, one can find that the geodesic $C_{0}$
with fixed endpoints is stable and locally minimizes the arc length in a
negatively curved Riemannian manifold. If we consider the perturbation by
varying the geodesic from one to another one, which means that we only focus
on a one-parameter family of neighboring geodesic $C_{z}$. Only special
variation vector $\mathbf{V}$ can vary one geodesic to another one. The
restriction is nothing but the geodesic deviation equation, i.e.,

$\frac{D^{2}\mathbf{V}}{\partial
s^{2}}+R\left(\mathbf{V},\mathbf{T}\right)\mathbf{T}=0\,,$ (4.10)

which is a group of second order linear ordinary differential equations for
$\mathbf{V}$. In differential geometry, the geodesic deviation equation is
also known as Jacobi equation, whose solution is called the Jacobi field. The
variation of arc length of geodesic $C_{0}$ generated by Jacobi field
$\mathbf{v}$ is thus given by

$\begin{split}L^{\prime\prime}(C_{0})&=\frac{1}{L_{0}}\left(g\left(\mathbf{t},\nabla_{\mathbf{v}}\mathbf{v}\right)+g\left(\mathbf{v},\nabla_{\mathbf{t}}\mathbf{v}\right)\right)\bigg{|}^{1}_{0}\,,\\\
\end{split}$ (4.11)

where the contributions only come from the initial and end points like the
first variation for geodesic.

Figure 15: The extremal geodesic (denoted by the blue curve) starting from the
boundary at a fixed point $P$ in Poincaré AdS spacetime intersects the dS
brane, which is shown by the gray circle. If the geodesic intersects the dS
brane from the inside (i.e., at the yellow point $Q_{2}$ in this figure), the
extrinsic curvature is positive, indicating that the extremal geodesic is
unstable at this intersection.

By using necessary ingredients reviewed before, we move to analyzing the
stability of the extremal surfaces anchored on a brane living in (Euclidean)
AdS bulk spacetime, as illustrated in figure 14. First of all, we are only
interested in the cases with one endpoint fixed at $s=0$ but non-vanishing
variation at another endpoint $s=1$ on the brane. The variation of the
geodesic is realized by varying the endpoint on the brane, which indicates
that the variation vector $\mathbf{V}$ is nothing but the tangent vector along
the brane. Since we assume that the original geodesic is extremal with respect
to the brane profile, the first variation of the geodesic should simply
vanishes, implying $g\left(\mathbf{t},\mathbf{v}\right)=0$ on the brane. It
gives rise to a fact that the tangent vector $\mathbf{t}$ of the geodesic is
related to the unit normal vector $\mathbf{n}$ of the brane, namely

$\mathbf{n}=\frac{1}{L_{0}}\mathbf{t}\big{|}_{\rm{brane}}\,,\qquad\text{with}\quad||\mathbf{n}||=1\,.$
(4.12)

We can thus show that the boundary term in the second variation formula (4.7)
is related to the second fundamental form $K_{\mu\nu}$ of the brane because of

$\frac{1}{L_{0}}g\left(\mathbf{t},\nabla_{\mathbf{v}}\mathbf{v}\right)\big{|}_{\rm{brane}}=g\left(\mathbf{n},\nabla_{\mathbf{v}}\mathbf{v}\right)=-g\left(\nabla_{\mathbf{v}}\mathbf{n},\mathbf{v}\right)=-K(\mathbf{v},\mathbf{v})\,.$
(4.13)

Also noting that the bulk Riemannian tensor term in the integral is related to
the sectional curvature161616The sectional curvature in our convention is
defined by  $K_{\rm{set}}(\mathbf{X},\mathbf{Y})\equiv\frac{\langle
R(\mathbf{X},\mathbf{Y})\mathbf{Y},\mathbf{X}\rangle}{\langle\mathbf{X},\mathbf{X}\rangle\langle\mathbf{Y},\mathbf{Y}\rangle-\langle\mathbf{X},\mathbf{Y}\rangle^{2}}=\frac{R(\mathbf{X},\mathbf{Y},\mathbf{X},\mathbf{Y})}{\langle\mathbf{X},\mathbf{X}\rangle\langle\mathbf{Y},\mathbf{Y}\rangle-\langle\mathbf{X},\mathbf{Y}\rangle^{2}}\,.$
(4.14)

$g\left(R(\mathbf{v},\mathbf{t})\mathbf{t},\mathbf{v}\right)=K_{\rm{set}}(\mathbf{v},\mathbf{t})||\mathbf{v}\wedge\mathbf{t}||^{2},$
(4.15)

Combining all previous expressions before, the variation generated by an
arbitrary orthogonal variation vector along the brane can be rewritten as

$\begin{split}L^{\prime\prime}(C_{0})&=-K(\mathbf{v},\mathbf{v})+\frac{1}{L_{0}}\int^{1}_{0}\left[||(\nabla_{\mathbf{t}}\mathbf{v})||^{2}-K_{\rm{set}}(\mathbf{v},\mathbf{t})||\mathbf{v}\wedge\mathbf{t}||^{2}\right]ds\,.\\\
\end{split}$ (4.16)

As the first result, we can conclude that the necessary and sufficient
condition for the extremal geodesic anchored on the brane to be stable is
given by requiring

$\begin{split}L^{\prime\prime}(C_{0})&=\left(-K(\mathbf{v},\mathbf{v})+g\left(\mathbf{v},\nabla_{\mathbf{n}}\mathbf{v}\right)\right)\big{|}_{\rm
brane}>0\,.\\\ \end{split}$ (4.17)

for any tangent vector $\mathbf{v}$ on the brane.

Furthermore, it is remarkable to notice that AdSd+1 or Euclidean AdS space has
constant negative sectional curvature, a characteristic feature of AdS spaces.
The value of its sectional curvature does not depend on the chosen plane or
point in the manifold, indicating uniformity in curvature throughout the
spacetime. Together with the fact that
$||(\nabla_{\mathbf{t}}\mathbf{v})||^{2}$ and
$||\mathbf{v}\wedge\mathbf{t}||^{2}$ are positive, we can find that the
integral term appearing in the second variation is always positive. As a
result, it is obvious that the second variation is bounded below by the
extrinsic curvature of the brane. If the brane has a negative extrinsic
curvature everywhere, i.e., 171717Note that this is only a sufficient but not
necessary condition since the integral term is always non-negative.

$L^{\prime\prime}(C_{0})\geq-K(\mathbf{v},\mathbf{v})\big{|}_{\rm brane}>0\,,$
(4.18)

one can simply claim that the geodesic orthogonal to the brane is stable, that
is, locally minimizes the arc length. In other words, the only term that could
spoil the stability of the geodesics anchored on the brane is the positive
extrinsic curvature.

In the last part of this subsection, we shall now apply the aforementioned
results to the EOW brane in AdS space, which satisfies the Neumann boundary
condition (3.3), as discussed in the preceding sections. As a consequence of
the Neumann boundary condition, it is straightforward to demonstrate that the
extrinsic curvature of the corresponding d-dimensional brane is determined by
its tension, i.e.,

$K=\frac{d}{d-1}\mathcal{T}\,,\qquad K_{ij}=\frac{\mathcal{T}}{d-1}g_{ij}\,.$
(4.19)

It is obvious that the sufficient condition (4.18) can be consistently
satisfied when the brane with positive tension is considered. In the case of
negative tension, it is a simple matter to verify that the extremal geodesic
anchored on the dS brane is inherently unstable, exhibiting a locally maximal
length along the dS brane. In contrast, the extremal surface anchored on the
AdS brane remains stable. This distinction can be attributed to the fact that
the extrinsic curvature of the AdS brane is always relatively minor. More
explicitly, we can get
$K=\frac{d}{d-1}\mathcal{T}\leq\frac{d}{L_{\textrm{\tiny AdS}}}$ for a AdS
brane since it can only exist with a lower tension
$|T|<\frac{d-1}{L_{\textrm{\tiny AdS}}}$ Kawamoto:2023wzj . Consequently, it
can be demonstrated that the sufficient and necessary condition derived in eq.
(4.17) will always be satisfied for the AdS brane, regardless of the sign of
its extrinsic curvature. In contrast, the dS brane, which satisfies the
Neumann boundary condition, is supported by a large tension, with
$|T|>\frac{d-1}{L_{\textrm{\tiny AdS}}}$. In consequence, the dS brane with
positive tension will invariably satisfy

$K(\mathbf{v},\mathbf{v})|_{\rm{dS\,\,brane}}\propto
K|_{\rm{dS\,\,brane}}>\frac{d}{L_{\textrm{\tiny AdS}}}\,,$ (4.20)

resulting in the violation of the condition given by eq. (4.17).

In conclusion, the non-stable geodesic on the dS brane can be attributed to
the positive sign and the magnitude of its extrinsic curvature. As illustrated
in figure 15, the extremal geodesic is stable outside the dS bubble, where
$K<0$, and is non-stable inside the dS bubble, where $K>0$.

###  4.2 Instability of extremal surfaces in AdS/BCFT

Figure 16: The setup for computing the holographic entanglement entropy in
AdS/BCFT.

To elucidate the conclusion from the geodesic variation with more concrete
examples, let us examine the AdS3/BCFT2 setup, where the spacetime is
described by the Poincaré AdS3 metric, as follows:

$ds^{2}=\frac{-dt^{2}+dx^{2}+dz^{2}}{z^{2}}\,.$ (4.21)

We assume that the physical space is bounded by a static two dimensional EOW
brane. Assuming the worldsheet of the EOW brane is symmetric in time at $t=0$,
we characterize its profile at this instant as $z=Z(x)$. In the subsequent
analysis, we concentrate on the holographic entanglement entropy at $t=0$,
which allows us to confine the three-dimensional space to the two-dimensional
slice delineated by the domain $0<z\leq Z(x)$. This region is illustrated in
blue in figure 16.

The dual BCFT resides on the half line $x\geq 0$, extended in the trivial time
direction. We are interested in the holographic entanglement entropy
$S_{\mathcal{A}}$ for the interval $\mathcal{A}=[0,\xi]$ at $t=0$. The
intersection of the geodesic $\Gamma$ emanating from $x=\xi$ with the EOW
brane is characterized by the coordinates $(x,z)=(a,Z(a))$. According to the
standard holographic prescription for entanglement entropy in the AdS/BCFT
correspondence, the value of $a$ is determined by minimizing the geodesic
length $D$ with respect to the parameter $a$ Takayanagi:2011zk ; Fujita:2011fp
.

The geodesic distance $D$ between $(\xi,\epsilon)$ and $(a,Z(a))$ is defined
by

$\cosh
D=\frac{(Z(a)-\epsilon)^{2}+(\xi-a)^{2}}{2Z(a)\epsilon}\equiv\mathcal{L}\,,$
(4.22)

where $z=\epsilon$ is the standard UV regulator. By requiring that the
derivative of $D$, or equivalently $\mathcal{L}$, with respect to $a$
vanishes, we find

$\xi-a=-\frac{Z(a)}{\dot{Z}(a)}+\sqrt{\frac{Z(a)^{2}}{\dot{Z}(a)^{2}}+Z(a)^{2}-\epsilon^{2}}\,,$
(4.23)

where $\dot{Z}=\frac{dZ(a)}{da}$. This determines the value of $a$ as a
function of $\xi$. In the limit $\epsilon\to 0$, which corresponds to the BCFT
living at the AdS boundary, we proceed with the analysis.

Now, we inquire whether the saddle point is stable, that is, whether it
represents a local minimum. To determine this, we need to compute the second-
order derivative,

$\frac{d^{2}\mathcal{L}}{da^{2}}=\frac{2Z^{2}}{\dot{Z}^{2}}\left(\dot{Z}^{2}+\dot{Z}^{4}-Z\ddot{Z}+Z\ddot{Z}\sqrt{1+\dot{Z}^{2}}\right)\,.$
(4.24)

From this, it is evident that for the saddle point to become unstable, i.e., ,
for $\frac{d^{2}\mathcal{L}}{da^{2}}<0$, it is necessary to have $\ddot{Z}<0$.
This condition is indeed generally possible. The null energy condition
stipulates that

$\ddot{Z}+(1+\dot{Z}^{2})\partial^{2}_{t}Z\leq 0\,,$ (4.25)

where $\partial_{t}$ denotes the derivative with respect to the time variable,
assuming the time symmetry condition $Z(x,t)=Z(x,-t)$. Specifically, for a
static EOW brane, denoted by $z=Z(x)$, the time derivative is absent, and this
condition simplifies to $\ddot{Z}\leq 0$.

Figure 17: Left: The geodesic function $\mathcal{L}(a)$ with setting
$\xi=0,\epsilon=1$ is illustrated as a function of the intersection point $a$
on the EOW brane, which is parameterized by eq. (4.26). Right: The extrinsic
curvature component, defined as $K(\mathbf{v},\mathbf{v})$, is plotted as a
function of $a$.

For instance, consider the profile of an EOW brane described by

$z=Z(x)=p\sqrt{1-x^{2}},,$ (4.26)

where $p=1$ corresponds to the conventional constant tension brane. It is
evident that $\frac{d^{2}\mathcal{L}}{da^{2}}\propto 2p^{2}-p^{4}$ at
$a=\xi=0$. For $p=2$, the geodesic distance between $(0,0)$ and $(a,Z(a))$ is
depicted in figure 17. It is observed that the saddle point at $a=0$ is
unstable, whereas there are two stable saddle points. In the AdS/BCFT
framework, where the brane extends within the region $-1\leq x\leq 1$ and
intersects the AdS boundary at $z=0$, the stable saddle points yield the
correct holographic entanglement entropy, and the corresponding geodesic is
extremal and locally minimal. However, if the EOW brane is confined to a
smaller range, for example, $-0.5<x<0.5$, there exists no extremal point that
minimizes the length. This situation is analogous to our non-extremal island
for the CFT on a half dS2 coupled to gravity on a half dS2. It is noteworthy
that this phenomenon is not exclusive to positive curvature on the EOW brane.
To illustrate this, let us consider the EOW brane as static. The scalar
curvature on the EOW brane is then given by

$R=-\frac{2(\dot{Z}^{2}+\dot{Z}^{4}-Z\ddot{Z})}{(1+\dot{Z}^{2})^{2}}\,.$
(4.27)

Applying the null energy condition $\ddot{Z}<0$, we deduce that this brane
invariably exhibits a negative curvature $R\leq 0$.

On the other hand, the extrinsic curvature for the static EOW brane described
by the profile $z=Z(x)$ is given by

$K_{tt}=\frac{1}{Z^{2}\sqrt{\dot{Z}^{2}+1}},\ \
K_{xx}=-\frac{Z\ddot{Z}+\dot{Z}^{2}+1}{Z^{2}\sqrt{\dot{Z}^{2}+1}}.$ (4.28)

Additionally, the contraction with respect to the variation vector
$v^{\mu}=P^{\mu}_{x}$ ($P_{\nu}^{\mu}$ being the projection tensor on the
brane) along the $x$ direction is given by

$K(\mathbf{v},\mathbf{v})=-\frac{Z^{2}(Z\ddot{Z}+\dot{Z}^{2}+1)}{\left(\dot{Z}^{2}+1\right)^{5/2}}\,.$
(4.29)

Here, we have chosen the outward-pointing unit normal vector of the brane.
Imposing the null energy condition $\ddot{Z}<0$, the quantity
$K(\mathbf{v},\mathbf{v})$ can be either positive or negative, and its
magnitude can be arbitrarily large in certain regions of $x$. Thus, in
general, there exist both stable and unstable saddle points, as analyzed in
section 4.1. We can then consider the example of the profile in eq. (4.26) and
plot the function $K(\mathbf{v},\mathbf{v})$ as a function of the intersection
point $a$ on the EOW brane, as shown in figure 17. We find that
$K(\mathbf{v},\mathbf{v})$ is always positive in this region, and the minimal
stable points appear in regions where $K(\mathbf{v},\mathbf{v})$ is small.
However, the maximal point appears when $K(\mathbf{v},\mathbf{v})$ is large,
rendering the second variation negative at that point.

Figure 18: Left: The renormalized geodesic distance $\bar{D}_{12}$ (4.33) as a
function of the endpoint located at $\theta=\theta_{2}$ on the brane. In the
case of a deformed dS brane with a negative value of $q$, there exists the
possibility of an extremal point that is locally minimal. Right: The extrinsic
curvature $K(\mathbf{v},\mathbf{v})$ as a function of $\theta_{2}$. The
negative value of the extrinsic curvature $K(\mathbf{v},\mathbf{v})$ provides
a sufficient condition for the extremal geodesic to be locally minimal. For
both plots, we choose $\eta_{b}=5$, $\theta_{1}=0=t_{1}$.

Next, we consider a setup where the EOW brane is a dS brane or its
deformation. Using the AdS/dS slicing of AdS3,

$ds^{2}=d\eta^{2}+\sinh^{2}{\eta}\quantity(-dt^{2}+\cosh^{2}{t}d\theta^{2})\,,$
(4.30)

the dS brane is located at the hypersurface $\eta=\eta_{b}$. We can generalize
this setup to consider a brane determined by a more generic profile
$\eta=f(\theta)$. We examine variations along the $\theta$ direction, where
the variation vector is $v^{\mu}=P^{\mu}_{\theta}$, with $P^{\mu}_{\nu}$ being
the projection tensor on the brane. The contraction of the extrinsic curvature
with the variation vectors, i.e., $K(\mathbf{v},\mathbf{v})$ is given by

$\frac{\cosh^{2}t\sinh f\sqrt{f^{\prime 2}\sech^{2}t\csch^{2}f+1}\left(\cosh
f\left(2f^{\prime 2}+\cosh^{2}t\sinh^{2}f\right)-f^{\prime\prime}\sinh
f\right)}{\left(f^{\prime 2}+\cosh^{2}t\sinh^{2}f\right)^{3}}\,.$ (4.31)

While the denominator of $K(\mathbf{v},\mathbf{v})$ is always positive, the
numerator can be either positive or negative and is time-dependent. For the dS
brane where $f(\theta)=\eta_{b}$, we have
$K(\mathbf{v},\mathbf{v})=\coth\eta_{b},\csch^{2}\eta_{b}\sech^{2}t$, which
retains the sign of $\eta_{b}$. A positive $\eta_{b}$ may indicate instability
of the extremal surface. To explore the deformed dS brane, we consider
modifying $f(\theta)$, such as

$f(\theta)=\eta_{b}-q\cos\theta\,,$ (4.32)

with $\theta\in[\frac{\pi}{2},\frac{3\pi}{2}]$. For positive $q$, the brane is
deformed towards the boundary, and $K(\mathbf{v},\mathbf{v})$ remains
positive, potentially leading to instability. Conversely, for negative $q$,
the brane contracts inwards, which can result in $K(\mathbf{v},\mathbf{v})$
being negative in certain regions, thus ensuring that the geodesic remains
minimal under variations in the spatial direction.

On the other hand, we can directly examine the variation of the geodesic
length. The geodesic distance $D_{12}$ between one endpoint
$(\eta_{\infty},t_{1},\theta_{1})$ on the asymptotic boundary and the other
endpoint $(f(\theta_{2}),t_{2},\theta_{2})$ on the EOW brane is given by eq.
(3.6). As $\eta_{\infty}\to\infty$, the regularized geodesic length is

$\bar{D}_{12}=\log(\cosh{f(\theta_{2})}-\sinh{f(\theta_{2})}\quantity(\cosh{t_{1}}\cosh{t_{2}}\cos{(\theta_{1}-\theta_{2})}-\sinh{t_{1}}\sinh{t_{2}}))\,.$
(4.33)

We first consider the extremal condition, which requires that the first
derivatives with respect to both $t_{2}$ and $\theta_{2}$ vanish. The
extremization condition for the time direction simplifies to

$t_{2}=\text{arctanh}\frac{\tanh t_{1}}{\cos(\theta_{1}-\theta_{2})}\,,$
(4.34)

while the condition for the spatial direction generally yields a
transcendental equation, which is difficult to solve analytically. To
facilitate the calculation, we examine the example defined in eq. (4.32) to
illustrate the dependence of the geodesic length on $\theta_{2}$. For
comparison, the corresponding $K(\mathbf{v},\mathbf{v})$ is shown in figure
18. It is evident that negative $K(\mathbf{v},\mathbf{v})$ provides a
sufficient condition for a positive $L^{\prime\prime}(C_{0})$, which indicates
a stable extremal geodesic. This is particularly evident in the third case,
where $q$ is negative and of a large magnitude.

##  5 Conclusions and Discussions

In this paper, we explore the calculation of entanglement entropy in setups
where a CFT is defined on one half of a two-dimensional de Sitter space, while
a gravitational theory is defined on the other half across a timelike
boundary. Considering that the subsystem resides within the CFT region, we
initially applied the island formula. However, we discovered that no extremal
configuration of the island minimizes the entropy functional, as discussed in
section 2. We found an extremal island solution that results in a local
maximum, which is physically inadmissible. This situation contrasts with cases
involving CFTs coupled to black holes in AdS spaces, which have been
extensively studied in recent years. Nevertheless, in more general spacetimes,
both extrinsic and intrinsic curvatures in higher-dimensional spaces play
significant roles, as outlined in section 4. Consequently, these phenomena are
not limited to specific de Sitter spacetimes but may occur in a wide range of
gravitational settings.

To address issues related to the extremal island in dS space, section 3
explores the higher dimensional gravity dual through the framework of double
holography. By computing the holographic correlation functions, specifically
those of twist operators relevant to entanglement entropy calculations, in the
AdS/BCFT setup involving the EOW brane, we found that the correction
computation of the entanglement entropy is provided by a “non-extremal
island”. This implies that the chosen island does not extremize the entropy
functional but rather globally minimizes it. We further corroborated this
approach through a direct holographic entanglement entropy calculation using
the geodesic (denoted by $\Gamma_{\mathcal{A}}$ in the AdS3/CFT2 setup).
Utilizing this prescription, we computed the entanglement entropy in a
Euclidean setup where a CFT on a two-dimensional hemisphere is coupled to a
gravitational theory on the other hemisphere. We observed that for a small
subsystem, the geodesic remains connected, resulting in the entanglement wedge
excluding any portion of the gravitational region. As the size of the
subsystem increases, a phase transition occurs, and the geodesic becomes
disconnected. In contrast to standard scenarios where gravity resides in non-
positive curvature spaces, we found that in our de Sitter setup, the geodesic
always terminates at the boundary of the half dS space. Consequently, in this
island phase, the entanglement wedge encompasses the entire gravitating
region.

In the Lorentzian setup, where a CFT on one half of dS space is coupled to a
gravitational theory on the other half, we can compute the entanglement
entropy by analytically continuing the Euclidean results. We demonstrated that
this method is effective in the early stages of time evolution. This finding
was supported by explicitly computing the geodesic lengths in Lorentzian
geometry according to the doubly holographic description. However, at late
times, we observed that the endpoints of the geodesics on the EOW branes reach
the future infinity of the timelike boundary. Beyond this critical point, no
physically sensible geodesics exist in Lorentzian spacetime. Consequently, the
analytically continued entropy becomes complex-valued, akin to the pseudo
entropy Nakata:2020luh , indicating that the geodesic corresponding to this
entropy extends into a complexified coordinate direction. Additionally,
complex entanglement entropy may arise from regulating horizon-like structures
or quasi-normal modes Kawamoto:2023ade . A more comprehensive understanding of
the late-time behavior remains an open question. Although this paper focuses
on two-dimensional examples for simplicity, we anticipate that our findings
can be generalized to higher dimensions in a qualitatively similar manner.

Finally, it is intriguing to explore quantum corrections to the non-extremal
island contributions within the doubly holographic framework. In the leading
classical gravity approximation, the standard area formula
$\frac{\rm{Area}(\Gamma)}{4G_{\textrm{\tiny N}}}$ applies. Typically, in
holographic entanglement entropy, gravity loop corrections are expressed as
$\sum_{n=0}^{\infty}\mathcal{O}\left((G_{\textrm{\tiny N}})^{n}\right)$.
However, in the context of non-extremal islands, this conventional approach
seems inadequate, as the leading contribution does not satisfy the equation of
motion and does not represent the full saddle point. Consequently, the next
leading contributions could surpass the order of $(G_{\textrm{\tiny N}})^{0}$.
Addressing this issue and gaining a deeper understanding of the implications
of quantum corrections remains an important challenge for future research.

###### Acknowledgements.

We would like to thank Vijay Balasubramanian, Wei Song, Yu-ki Suzuki, Tomonori
Ugajin, Zixia Wei and Yuzhen Zhang for useful comments and discussion. This
work is supported by MEXT KAKENHI Grant-in-Aid for Transformative Research
Areas (A) through the “Extreme Universe” collaboration: Grant Number 21H05187.
TT is also supported by Inamori Research Institute for Science and by JSPS
Grant-in-Aid for Scientific Research (A) No. 21H04469. TK is supported by
Grant-in-Aid for JSPS Fellows No. 23KJ1315.

##  Appendix A Islands in dS JT gravity

Figure 19: The Penrose diagram of the dS2 black hole spacetime, where the
dilaton profile is given by eq. (A.3).

This paper’s primary focus is on the issues pertaining to the antipodal island
in dS gravity and its resolution. To this end, our result from holographic
correlation functions suggests the non-extremal island, situated at the edge
of the gravitational dS space. In order to align with the doubly holographic
model in AdS3, the two-dimensional dS gravity under consideration is selected
to be the Liouville gravity. One might inquire whether the unphysical nature
of the antipodal island is a consequence of the fact that induced dS2 gravity
is not dynamical. However, this appendix will demonstrate that it is, in fact,
a universal feature of the cosmological horizon by considering dS JT gravity.
Additionally, it is possible that there exists a physical extremal island
whose boundary is given by the maximin surface. However, the extremal island
must be situated in proximity to the black hole singularity.

###  A.1 dS2 black hole spacetime

As we have noted at the beginning of section 2, there are two types of dS JT
gravity Maldacena:2019cbz ; Cotler:2019nbi ; Sybesma:2020fxg ; Kames-
King:2021etp ; Svesko:2022txo , depending on whether there is a topological
term originating from the higher dimensional extremal black hole. The island
formula has been studied in dS JT gravity by many paper from different
perspectives, see e.g., Chen:2020tes ; Hartman:2020khs ;
Balasubramanian:2020xqf ; Sybesma:2020fxg ; Aguilar-Gutierrez:2021bns ; Kames-
King:2021etp ; Aalsma:2021bit ; Teresi:2021qff ; Levine:2022wos ;
Aalsma:2022swk ; Chang:2023gkt ; Balasubramanian:2023xyd . We start from
considering the JT gravity derived from the Nariai limit of four dimensional
Schwarzschild-de Sitter black hole:

$S_{\rm JT^{\prime}}=\frac{\Phi_{0}}{16\pi G_{\textrm{\tiny N}}^{(2)}}\int
d^{2}x\sqrt{-g}R+\frac{1}{16\pi G_{\textrm{\tiny N}}^{(2)}}\int
d^{2}x\sqrt{-g}\,\Phi\left(R-\frac{2}{L^{2}}\right)+\text{boundary terms}\,,$
(A.1)

where the first $\Phi_{0}$ term with a proper boundary terms is topological
for two dimensional spacetime but would be important for the following
discussion. It is straightforward to get the equation of motion

$R=\frac{2}{L^{2}}\,,\qquad\left(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu}+\frac{g_{\mu\nu}}{L^{2}}\right)\Phi=0\,.$
(A.2)

where the stress tensor of matter field is fixed as zero. For the solution
with non-vanishing expectation values of the stress tensor, refer to e.g.,
Balasubramanian:2020xqf . The solution of dS2 spacetime and dilaton field in
conformal coordinates reads181818The solutions in other coordinates can be
derived from conformal coordinates by coordinate transformations. For example,
the dilaton profile is given by  $\Phi=\Phi_{r}\cos\theta\cosh
t=\Phi_{r}\,r=\Phi_{r}\frac{1+UV}{1-UV}\,,$ (A.3) in global coordinates,
static patch coordinates and Kruskal coordinates, respectively.

$ds^{2}=\frac{L^{2}}{\cos^{2}T}\left(-dT^{2}+d\theta^{2}\right)\,,\qquad\Phi(T,\theta)=\Phi_{r}\frac{\cos\theta}{\cos
T}\,.$ (A.4)

The corresponding Penrose diagram of dS2 spacetime is shown in figure 19,
where we have both a cosmological horizon and a black hole horizon. This two-
dimensional spacetime serves as a toy model for dS Schwarzschild black hole.

Although the extra topological term $\Phi_{0}$ which origins from the higher
dimensional extremal black hole is not dynamical, it gives rise to the ground
state entropy. More importantly, it then allows the dynamical dilaton $\Phi$
to be negative. As presented by the profile in eq. (A.4), the dilaton value is
negative in the black hole region. Due to the constraint from the eﬀective
Newton’s constant, we need to require

Although the additional topological term $\Phi_{0}$, which originates from the
higher-dimensional extremal black hole is not dynamical, it nevertheless gives
rise to the ground state entropy. Furthermore, this allows for the possibility