Encoded information of mixed correlations: the views from one dimension higher

$\text{Mahdis Ghodrati }^{a}$

aAsia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea

e-mails<EMAIL_ADDRESS>
After reviewing the JT gravity, we discuss the four saddles in the mixed
correlation measures of black holes Hawking radiation in the setup of
geometric evaporation of [1]. By looking from $1d$ higher point of view and
partial dimensional reduction, we examine the phase structures and the
universalities for these four saddles. We also discuss the behavior of quantum
error correction codes for each of these four phases, reaching to consistent
results. Then, instead of dimension reduction between Einstein gravity and JT,
we try to explore the connections between partition functions and saddles of
$3d$ Chern-Simons and $2d$ BF theories, $2d$ Liouville and $2d$ Wess-Zumino-
Witten models, and also the dimensionally reduced $1d$ Schwarzian and $1d$
particles on group. We comment on the connections between these theories in
the setup of mixed correlations and Page curves. We then examine further the
information that can be gathered by looking from $1d$ higher, specifically
through studying the negativity and Kaluza-Klein, charged, and extremal black
holes. Finally, we explore several measurable observables in the setup of
dimension reductions, such as quantum fluctuations, Kondo effect, Berry phase,
quantum focussing and complexity. We comment on how Berry phase could
determine the connectivity of spacetime and the types of replica wormholes,
and also sketch the connections between brane tomography model, inception
geometry and replica wormholes using the island formulation and mixed
correlation measures.

###### Contents

  1. 1 Introduction
  2. 2 The setup
  3. 3 Phase diagrams from different correlation measures
  4. 4 Entanglement structure before and after the Page time: the view from $1d$ higher
    1. 4.1 Before the Page time
    2. 4.2 After the Page time
    3. 4.3 Comparing with results from partial transpose
    4. 4.4 Quantum error corrections of black hole from $1d$ higher point of view
  5. 5 Comparing partition functions and density states of $3d$ versus $2d$
    1. 5.1 The $3d$ Chern-Simons gravity to $1d$ particle on group
    2. 5.2 The connections between JT model and Liouville gravity
    3. 5.3 The $3d$ Wess-Zumino-Witten model
  6. 6 Mixed correlations and Islands: the view from $1d$ higher
    1. 6.1 The view of negativity and island from $1d$ higher
    2. 6.2 Islands in Kaluza-Klein black holes
    3. 6.3 Islands for charged dilaton black holes and its view from higher $d$
    4. 6.4 Extremality from higher $d$
  7. 7 Measurable observables from $1d$ higher
    1. 7.1 Quantum Fluctuations
    2. 7.2 Kondo effect and island formulation
    3. 7.3 Black hole hair from higher $d$
    4. 7.4 Connectivity of spacetimes
    5. 7.5 Black hole secret sharing and Kaluza-Klein reduction
    6. 7.6 Quantum focussing conjecture in $2d$ from $3d$ point of view
    7. 7.7 Connections with complexity
    8. 7.8 Connections with other works
  8. 8 Conclusion

##  1 Introduction

Recently, the idea of looking the evolution of black holes and its phase
structures, in the setup of quantum gravity, from higher dimensions to lower
dimensions has been proved to be very fruitful. This general idea of looking
at black hole evaporation from one dimensional higher and specifically the
case of $2d$ Jackiw–Teitelboim (JT) gravity from $3d$ Einstein gravity model,
has been started in [2], while the effects of boosted quantum corrections on
quantum extremal surface (QESs) and the connections with the Page curve of
black holes were first studied in [3]. Related to this idea, the evolution of
the black holes has also been studied by the island/$\text{BCFT}_{2}$ setup in
various works which we mention throughout the work here.

We specifically want to extend this method of looking the evaporation of black
holes from $3d$ to $2d$ JT gravity, by using mixed correlation measures, and
partial, geometric dimension reduction used in [1]. After reviewing JT gravity
in section 2, in section 3, we comment in more details on the universalities
noted between the phase structures coming from various correlation measures,
that we observed in our previous work [4].

Instead of one angular interval, in our section 4, we consider two interval
subsystems, and then using mutual information and the critical bath size, we
explore the phase structure of Hawking radiation, for the case before and
after the Page time. We find there, many universalities between phases, coming
from different correlation measures and models of saddles of Hawking
radiation.

On the other hand, there are several formulations for holographic bulk
reconstruction such as tensor network, HKLL, quantum recovery channel, or
modular flow, where in [5], the connections between some of them have been
assessed. The boundary information data which reconstruct the bulk, can in
principle specify the phases or saddles as well. However, the transitions
between these saddles in the setup of black holes would be subtle. Viewing
this problem from one dimension higher and then using dimension reduction
would shed light on several insighting issues, which is the aim of this work.
In section 4.4, we apply the inequalities coming from quantum error
corrections (QEC), between these four saddles, where we find consistent
results for the possible ways to move between these saddles, with the results
of QEC.

The initial idea of geometric evaporation and dimension reduction in [1] have
been applied between $3d$ Einstein gravity and $2d$ JT model. This idea then
can be extended to other interesting and important quantum gravity toy models,
such as $3d$ Chern-Simons and $2d$ BF gauged theory, and also for the boundary
$2d$ Liouville or $2d$ Wess-Zumino-Witten (WZW) gauged models. Then, one can
go even one dimension lower and study the problem for the case of $1d$
Schwarzian or $1d$ particle on group model. The picture in figure 18, showing
the connections between these models would be very illuminative in our study
there. In our section 5, we try to sketch how the information is encoded
between various dimensions using these models, and the effects of the
parameters of partition functions on the behavior of Page curve in these
models.

This idea of looking from $1d$ higher and then dimensional reduction can be
applied for other interesting examples such as examining the behavior of
negativity in the presence of islands. The behavior of islands with
connections to Kaluza-Klein (KK) charge, dilaton charge and extremality will
be addressed in our section 6. There, we also comment on the dynamics of the
island inside of the black hole before and after the Page time, and the links
between its velocity with the charge, central charge and KK modes. We also
comment on the issue of finite entropy of extremal black holes using the
perspective of island formulation and looking from $1d$ higher point of view.

Later, in section 7, we push this idea further, to check the connections
between the behavior of black hole’s quantum information with other physical
measurable observables, such as measurable quantum fluctuations, connections
between island formulation and Kondo effect in condensed matter systems, black
hole hairs, quantum secret sharings, quantum focusing conjecture and
computational complexity. There, we specifically comment on the connections
between Berry phase and connectivity of spacetimes, and also connections
between three interesting models of replica wormholes, inception geometry and
brane tomography, in the setup of island formulation and dimensional
reduction. We also sketch how this idea is related to many other recent works.
Finally, we end with a short conclusion in section 8.

##  2 The setup

Most of the discussion of black hole evaporation and information loss used the
Jackiw–Teitelboim (JT) gravity toy model which could be written as

$\displaystyle I_{JT}=\frac{1}{16\pi G}\int
d^{2}x\sqrt{-g}\left[\phi_{0}R+\phi\left(R+\frac{2}{L^{2}}\right)\right]+\frac{1}{8\pi
G}\int dt\sqrt{-\gamma}\left[\phi_{0}K+\phi\left(K-\frac{1}{L}\right)\right].$

This gravity model can in fact, be obtained by the spherical reduction of the
$4d$ Einstein-Maxwell action,

$\displaystyle I_{EM}=\frac{1}{16\pi G_{4}}\int
d^{4}x\sqrt{-\hat{g}}\left(\hat{R}-\frac{1}{4\pi}\hat{F}^{2}\right)+\frac{1}{8\pi
G_{4}}\int d^{3}x\sqrt{\hat{\gamma}}\hat{K},$ (2.1)

Here he equation of motion for the dilaton would be
$\phi(r)=\frac{r}{L}\phi_{r}$.

Generally for simulating the black hole degrees of freedom, an additional term
in the from of an end-of-the-world (EOW) brane can be added to the JT action.

So with the ansatz

$\displaystyle
ds_{4}^{2}=g_{\mu\nu}(t,r)dx^{\mu}dx^{\nu}+(\phi_{0}+\phi(r,t))d\Omega_{2}^{2},$
(2.2)

in works such as [6], the action has been chosen as the combination of the JT
gravity with an EOW brane which has a tension $\mu$, written in the form of

$\displaystyle I=I_{JT}+\mu\int_{\text{brane}}dy,$ (2.3)

where the JT action would be written as

$\displaystyle
I_{JT}=-\frac{S_{0}}{2\pi}\left[\frac{1}{2}\int_{\mathcal{M}}R+\int_{\partial\mathcal{M}}K\right]-\left[\frac{1}{2}\int_{\mathcal{M}}\phi(R+2)+\int_{\partial\mathcal{M}}\phi
K\right],$ (2.4)

and $S_{0}$ is the zero temperature entropy of an eternal two-dimensional
black hole. In this model the main parameter is the number of orthonormal
states or flavors on the EOW brane denoted by $k$ which can actually model the
Hawking quanta falling inside the black holes. As $k$ increases, the later
regimes of the evaporating black hole could be probed. These states are also
entangled with the an auxiliary reference system $R$.

In the work of Verheijden and Verlinde [1], which is the base of our work
here, however, this EOW brane is replaced with the black hole, as the action
for matter $S_{\text{matter}}$, has been added to the JT last action, written
in the form of

$\displaystyle S=\frac{1}{16\pi G}\left[\int d^{2}x\sqrt{-g}\Phi_{0}R+\int
d^{2}x\sqrt{-g}\Phi\left(R+\frac{2}{\ell^{2}}\right)\right]+S_{\text{matter}},$
(2.5)

where here $\Phi_{0}$ is a constant. The first term which is topological,
determines the extremal entropy and after adding the appropriate boundary
term, it would give the Euler characteristic of the manifold, as it will be
shown next in the relation 2.6. The last term, $S_{\text{matter}}$, is some
arbitrary matter system which couples to the metric but not to the dilaton.

In the work of [7], the action for the JT gravity with the EOW brane has been
written as

$\displaystyle
I[\phi,g]=-S_{0}\chi-\frac{1}{4\pi}\Big{[}\int_{M}\sqrt{g}\phi(R+2)+\int_{\partial
M}\sqrt{h}\phi K\Big{]}+\phi_{r}\mu\int_{\text{ETW brane}}ds,$ (2.6)

where the role of Euler character and topology would be more clear, as $\chi$
is the Euler character of the Euclidean spacetime, $S_{0}$ is the extremal
entropy, $\mu$ is the tension of the EOW brane and $\phi$ is the dilaton. The
boundary condition for this model similar to Xi Dong’s case [6] is

$\displaystyle\text{Asymptotic AdS boundary:}\ \ \ \ \ \
\phi=\frac{\phi_{r}}{\epsilon},\ \ \ \ \
du^{2}\equiv\epsilon^{2}ds^{2}|_{bd},$ $\displaystyle\text{EOW brane
boundary:}\ \ \ \ \ \ \ n^{\alpha}\partial_{\alpha}\phi=\mu,\ \ \ \ \ K=0,$
(2.7)

As in Dong’s paper [6], the boundary would also be the Dirichlet condition on
an asymptotic boundary interval, while on the EOW branes, it would be Neumann
boundary conditions, as

$\displaystyle
ds^{2}\big{|}_{\partial\mathcal{M}}=\frac{1}{\epsilon^{2}}d\tau^{2},\ \ \ \
\phi=\frac{1}{\epsilon},\ \ \ \ \epsilon\to 0,$
$\displaystyle\partial_{n}\phi|_{\text{brane}}=\mu,\ \ \ \ \ K=0.$ (2.8)

Note that the nice feature of [1] is the presence of black hole instead of
EOW, which would let us to study correlations and various mixed entanglement
measures in a dynamical setup and study the connections between their
behaviors and Page curve, and also information loss paradox of black holes, as
we discuss in the next sections.

##  3 Phase diagrams from different correlation measures

Using the JT action as the toy model and its solutions, one could use several
different measures of entanglement and mixed correlation to study black hole
Hawking radiation. Then, we aim here to check the results coming from these
measures from $1d$ higher point of view.

In [6], the phase diagram of mixed system using “entanglement negativity” has
been found as shown in figure 1, which we briefly discussed in [4] and
commented on its connections with the phase structures of confining models,
and here we aim to study them in further details.

Note that in the calculation of the partition function for deriving the
entanglement negativity, four different saddles have been found corresponding
to four different types of pemutations, where the permutation $g=X$ which
corresponds to cyclic phase is related to the mixed correlation phase
discussed in [8]. The permutation $g=X^{-1}$ which corresponds to anti-cyclic
phase is related to the total correlation phase of [8]. The permutation
$g=\tau$ which corresponds to pairwise phase is related to the
$E_{Q}$-discontinous phase of [8], and the permutation $g=\mathds{1}$ which
corresponds to disconnected phase is related to the $E_{P}$ phase discussed in
[8]. We could connect these saddles with the various phases for two strips in
the confining geometries found in [9, 10]. Note that in figure 1, $k_{2}$ and
$k_{1}$ are the number of orthonormal states or flavors in each subsystem.

Then, the Rényi generalization of negativity could be written as

$\displaystyle
N_{k}(\rho_{AB})=\text{tr}\left((\rho_{AB}^{T_{B}})^{k}\right),$ (3.1)

where $N_{k}$ is the k-th Rényi negativity. This quantity in [11] has written
as

$\displaystyle N_{k}=\text{tr}\left[\rho_{AB}^{\otimes
k}\big{(}P_{A}(X)\otimes P_{B}(X^{-1})\big{)}\right],$ (3.2)

where $X$ is a k-cycle and $X^{-1}$ is its inverse, and $P_{M}(g)$ in general
is the representation of a permutation group element $g\in S_{k}$ on the $k$
copies of some subsystem $M$, and $P_{A}(X)$ and $P_{B}(X^{-1})$ are both
special cases of it. For the integer $k$, this relation could be written as
$N_{k}=\frac{Z_{k}}{Z_{1}^{k}}$, where $Z_{k}$ is the boundary partition
function on a $k$-fold branched over $M_{k}^{A,B}$ which can be obtained by
gluing $k$ copies of the original boundary spacetime $M_{1}$ cyclically along
$A$ and anti-cyclically along $B$.

Using the holographic duality, $Z_{k}$ can be calculated as

$\displaystyle Z_{k}=e^{-I[\mathcal{B}_{k}]},$ (3.3)

where $I[\mathcal{B}_{k}]$ is the on-shell action of proper bulk saddle point
solution $\mathcal{B}_{k}$, where its asymptotic boundary is the $k$-fold
cover $M_{k}^{A,B}$.

For the case of JT gravity with EOW brane with $k$ orthonormal states or
flavors, which are being splitted into two subsystems consisting of $k_{1}$
and $k_{2}$ states, such that $k=k_{1}k_{2}$, the pairwise connected geometry
would satisfy the relation $k_{1}k_{2}\gg e^{S_{0}}$ and $e^{-S_{0}}\ll
k_{1}/k_{2}\ll e^{S_{0}}$. These geometries correspond to the set of
permutations $\tau$ which are known as non-crossing pairings. An interesting
feature of these geometries is that for the even replica index $n$, a pairwise
connected geometry is constructed by connecting paired asymptotic boundaries
by two-boundary wormholes while for odd $n$, the geometries could be
constructed by the similar non-crossing pairings of the boundaries plus a
single one-boundary connected component and therefore these geometries would
spontaneously break the replica symmetry.

These geometries are dominant since $k_{1}$, $k_{2}$ and $e^{S_{0}}$ can be
put on the most equal footing by maximizing the sum of the three exponents in
the relation

$\displaystyle\text{Tr}[(\rho_{R}^{T_{2}})^{n}]\sim\frac{1}{(ke^{S_{0}})^{n}}\sum_{g\in
S_{n}}(e^{S_{0}})^{\chi(g)}\ k_{1}^{\chi(g^{-1}X)}\
k_{2}^{\chi(g^{-1}X^{-1})},$ (3.4)

and they contribute as

$\displaystyle
g=\tau\to\left(e^{S_{0}}\right)^{\lceil\frac{n}{2}\rceil}k^{\lfloor\frac{n}{2}\rfloor+1},$
(3.5)

where $\lceil\frac{n}{2}\rceil$ and $\lfloor\frac{n}{2}\rfloor$ are the
ceiling and floor functions.

So, the replica saddle points here can then be either symmetric or non-
symmetric. In [11], it has been shown that, in the phase where $I$ is positive
and entanglement wedge is connected, the nonsymmetric saddle points are
dominant in Rényi negativities. Therefore, most of states are in the pairwise
geometries which in fact break the replica symmetry there, and that is why in
works such as [6] for the JT gravity with the EOW brane and also in [10] for
the supergravity models, most of the space of the parameter regime are covered
by the pairwise case.

In general, a simple relation for logarithmic negativity could be written as

$\displaystyle E_{N}(\rho_{AB})\simeq\frac{1}{2}I(A:B)+\log\frac{8}{3\pi},$
(3.6)

which demonstrate the connections between negativity and mutual information as
we expected also from the phase structures.

Figure 1: Phase diagram of mixed system using negativity, from [6], and
comparing with plots of $D_{c}$ in confining geometries [10]. Figure 2: Phase
diagram of mixed system using wormholes [12], and comparing with the
configurations coming from mutual information and $D_{c}$ in confining
geometries. Figure 3: Phase diagram for confining geometries directly coming
from mutual information and $D_{c}$.

Then, in [12], the Page curves for “reflected entropy” have been presented
which is shown in figure 2, and the connections with the setup of confining
geometries are also shown. In that work this quantity has been studied in
random tensor network setup. Specifically the problem of the smoothing out of
the Page transition has been considered. In that work the phase diagram of the
single tripartite tensor for the Rényi reflected entropies as the function of
the bond dimension have also been presented which is very similar to the phase
diagrams found using the negativity in [6, 9] and mutual information and
critical distance, $D_{c}$ in [10]. For that case, the region where the
entanglement wedge is connected would correspond to $X_{A}+X_{B}>1$ and
$1-X_{B}<X_{A}<1+X_{B}$, which also corresponds to the pairwise connected
saddle for calculating negativity in JT gravity with EOW brane where
$k_{1}k_{2}\gg e^{S_{0}}$ and $e^{-S_{0}}\ll k_{1}/k_{2}\ll e^{S_{0}}$.

There, $X_{A}$ and $X_{B}$ are defined as

$\displaystyle X_{A}=\frac{\ln\chi_{A}}{\ln\chi_{C}},\ \ \ \ \
X_{B}=\frac{\ln\chi_{B}}{\ln\chi_{C}},$ (3.7)

where the $\chi_{i}$ corresponds to the bond dimension or the horizon areas of
the wormhole.

The Page phase transition is along the line $X_{A}+X_{B}=1$, and there are
other phase boundaries along $X_{A}=1+X_{B}$ and $X_{B}=1+X_{A}$ where the
derivative of the mutual information jumps, which in the QCD models would be
related to the confinement/deconfinement transitions, and chirality breaking
as explained in [10]. The bond dimensions then for the bath or radiation
system can be defined and the critical bath size would be connected to this
critical bond dimension, which is also related to the critical distance
$D_{c}$ studied in [10] for the confining geometries.

The actual phase diagram for confining geometries based on the different
scales of the setup, such as the distance between the two strips divided by
the AdS scale $X/R_{4}$, and width of strips $L$ divided by $R_{4}$, are shown
in figure 3. Note that in this phase diagram there are actually three scales,
the distance between the two equal strips $X$, the width of the strip $L$, and
the place of the hard wall in the confining models, $U_{0}$. These parameters
would correspond to the three constants of $k_{1}$, $k_{2}$ and $S_{0}$ in the
phase diagram derived in [6] and the three bond dimensions or the horizon area
of the three-boundary wormhole model of [12], and correspondingly to
$S_{A_{1}}^{\text{eq}}$, $S_{A}^{\text{(eq)}}$ and $S_{B}^{\text{(eq)}}$ of
[13, 14]. By tuning these parameters in each case the phase space can be
probed leading to the phase structures that have compatibilities with each
other.

In [13, 14] the entanglement negativity for the infinite and also finite
temperatures, using different methods, have been calculated and the phase
structures for the pattern of entanglement have also been constructed. For the
infinite temperature they found three phases, no entanglement (NE), maximally
entangled phase (ME), and entanglement saturation phase which are shown in
figure 4.

In that work for tracking the relative sizes of the subsystem, they used two
parameters defined as

$\displaystyle\lambda:=\frac{S_{A_{1}}^{\text{(eq)}}}{S_{A}^{\text{(eq)}}},\ \
\ \ \ \
c:=\frac{S_{A}^{\text{(eq)}}}{S_{A}^{\text{(eq)}}+S_{B}^{\text{(eq)}}}.$ (3.8)

We propose here that the maximally entangled phase (ME) for large values of
$c$, for the case of small $\lambda$ and big $\lambda$ of [13, 14] would be
distinct from each other, as the former one corresponds to anti-cyclic phase
of [6] and total correlation phase of [8], while the later corresponds to the
cyclic phase of [6] and total correlation phase of [8]. We also propose that
the entanglement saturation phase corresponds to the pairwise phase of [6] and
to the $E_{Q}$-discontinuous phase of [8]. In addition, the “no entanglement”
(NE) phase corresponds to the case of $E_{P}$-discontinuous phase of [8].

Figure 4: Phase diagram using negativity, found in [13, 14].

So the main point is that the similarities and correspondences that is being
observed in these phase diagrams point to the universalities in the
entanglement structures which shows itself in different measures of mixed
correlations and the resulting phase structures.

It should also be noted that the phase transitions between any two of these
phases are not sharp, but rather similar to black hole evaporation, in each
scenario first islands of other entanglement configurations would appear in
each phase region, which then would make that specific phase transition rather
smooth, but still the dominant saddle at each phase is the one that we
specified in each phase diagram here.

In fact in other works such as [12], it has been shown that for other measures
such as $(m,n)$-Rényi reflected entropy, there are also some significant non-
perturbative effects coming from new other saddles. This fact gives further
hints that in each phase with a specific dominant saddle, there are still
islands of other entanglement structures and saddles which could get
significant non-perturbative effects and then become dominant in different
limits, for instance, in the limit of large $m$ or $n$. So, this way, one
could find the corresponding “Page curve” for any of these specific measures.

One could use other probes of phase structures as well. For instance, in [15],
using the entanglement density of strip-subsystems, the entanglement structure
of systems where the $U(1)$ or translational symmetry is spontaneously broken
is studied. Specifically, there, for classifications and finding the
universalities, the area law or the first law of the entanglement entropy have
been used. In particular, using the $U(1)$ symmetry breaking pattern, the
normal or superconducting phases have been classified. Using one strip with
the size of $\ell$, it has been found that, in $(d+1)$ dimensions, the
entanglement entropy get a $\ell^{d}\log\ell$ contribution from the Fermi
surface, $\log(\ell^{d})$ from the Goldstone bosons and $\ell$-dependent
behavior from the topologically ordered degrees of freedom. The system size
$\ell$ would be related for instance to $k$ in the study of negativity of
black hole radiation.

Another interesting toy model for detecting black hole phases would be moving
mirrors [16], which can also point to the universalities mentioned before as
well. In [16], moving mirror models in $2d$ CFT have been studied where based
on their late time behaviors, they also found “four” classes, namely, the type
A or timelike mirrors, type B or escaping mirrors, type C or chasing mirrors
and type D or terminated mirrors, where each class has its own specific
characteristics for the energy stress tensor and entanglement entropy, which
essentially can be derived from inspecting the “endpoints” of these mirrors.
Also, the dual of these moving mirrors would be the end of the world branes
with different profile structures. We propose here that these “four”
categories of moving mirrors are essentially the four structures of mixed
correlation observed in confining backgrounds as in [9, 10] or the structures
of replica wormholes observed by studying the logarithmic negativity in [6].
Even in the dynamical setups of [17], for the localized shockwaves inside the
black hole, the four regimes have been observed and their dual quantum
circuits model have been constructed. Now the the exact correspondence between
each categories (and also the subcategories) of these models can be detected.

Based on the symmetry structures and boundary conditions, we expect that the
type A or the timelike mirror corresponds to the disconnected, $g=1$ or
$E_{P}$ discontinuous regime shown in figure 1. Then, the type B or the
escaping mirror would correspond to the Anti-cyclic, $g=X^{-1}$ case with the
total correlation regime. Type C or the chasing mirror would correspond to
Cyclic, $g=X$ with Mixed correlation regime and finally type D or terminated
mirror would correspond to the Pairwise, $g=\tau$ or $E_{Q}$ discontinuous
regime. The correspondences between the subcategories then similar to what
have been found in [13, 14] could also be associated to those found in [16].

By tracing the similarities between these models and categories, it could be
noticed that the case of black hole radiation where subregions are correlated
would be related to type B or escaping mirrors and therefore mostly have the
anti-cyclic $g=X^{-1}$ or the ”Total correlation” saddle. For the case where
particles get created and then evaporated, the process could be modeled by the
kink mirrors which is related to a subcategory of type A or timelike mirrors
which the essentially is the two disconnected wedges, with $g=1$ and $E_{P}$
discontinuous.

When the mirrors move very fast, i.e, $p^{\prime}(u_{\text{end}})=\infty$ and
$p(u_{\text{end}})=\infty$, creating the chasing mirror category, where the
degrees of freedom get accumulated, a connected wedge would be constructed,
and the cyclic mixed correlation saddle with $g=X$ would come up. Lastly, the
terminated mirror where the mirror trajectory would terminated at a particular
point in the bulk spacetime, the null points would appear which is related to
projection/preparation of direct product state, creating the Pairwise case
with two disconnected bulk wedges reaching to the end point of bulk with
$p(u_{\text{end}})=v_{\text{end}}$ and $p^{\prime}(u_{\text{end}})=\infty/0$.

Another evidence for all of the correspondences between the saddles we
mentioned above come from the analysis of the energy flux. As found in [16],
for type A and B, the energy flux would be finite and these two correspond to
cases where the bulk wedge does not reach to the end wall, and therefore they
are related to the disconnected and anti-cyclic saddles correspondingly.

It worths to mention here that, in [8], then the optimized correlation
measures in two-dimensional thermal states which are dual to spacetimes
containing black holes have been studied. These measures were EoP,
Q-correlation, R-correlation and squashed entanglement, and they probed the
parameter space for the phase diagrams. They proposed that the “Q-correlation”
would have the richest behavior, so it would be interesting to use this
measure in the setup of [1], to analyze the phase diagram of $2d$ spacetime
from the perspective of $3d$ BTZ.

##  4 Entanglement structure before and after the Page time: the view from
$1d$ higher

In this section the entanglement structures of radiation and bath in the two-
dimensional model of Verlinde and Verheijden (VV) [1] and for the cases before
and after the Page time are examined.

For the setup of VV, from the length of the geodesic, the entanglement entropy
before the Page time has been found as

$\displaystyle
S=\frac{1}{4G}\left(2\log\sinh\frac{\pi}{\beta}(2\pi\ell(1-\alpha)-2b)\right),$
(4.1)

and then after the Page time, the geodesic would “jump” and so the entropy
would become

$\displaystyle
S=\frac{1}{4G}\left(2\log\sinh\frac{\pi}{\beta}(2\pi\ell\alpha+2b)\right).$
(4.2)

Now instead of one interval, which corresponds to one angle $\alpha$ in the
relations 4.1 or 4.2, we take two intervals with angles $\mu$ and with
distance $\nu$ among them. Totally, we have the angular distance
$\alpha=2\mu+\nu$ for the whole angular interval. Then, we can use the mutual
information relation as

$\displaystyle I(A:B)=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (4.3)

which for our case would be $S_{A}=S_{B}=S|_{\mu}$, and
$S_{AB}=S(2\mu+\nu)+S(\nu)$. So the results for these two cases would be the
dynamical extension of the work of [14] to the setup of [1].

Using our diagrams, we will show that before the Page time, the critical size
of the bath $b_{c}$ where the phase transition happens is much bigger than
both the system size $\mu$ and $\nu$, and also their sum $\mu+\nu$, which
corresponds to the case where $q=\frac{L_{B}}{L_{A}}\gg 1$. After the Page
time, the critical size of bath would be very small, and therefore
$q=\frac{L_{B}}{L_{A}}\ll 1$.

###  4.1 Before the Page time

From the relation for the mutual information 4.3 and entanglement entropies
4.1 and 4.2, for the case before the Page time, we find

$\displaystyle\sinh^{2}\frac{\pi}{\beta}\Big{(}2\pi\ell(1-\mu)-2b_{c}\Big{)}=\sinh\frac{\pi}{\beta}\Big{(}2\pi\ell(1-\nu)-2b_{c}\Big{)}\sinh\frac{\pi}{\beta}\Big{(}2\pi\ell(1-2\mu-\nu)-2b_{c}\Big{)},$
(4.4)

which has four solutions as

$\displaystyle b_{c}(1)$
$\displaystyle\to\frac{\beta}{4\pi}\left(\log(2)-4i\pi
c_{1}-2\log\left(-\sqrt{\frac{e^{-\frac{4\pi^{2}l}{\beta}}\left(\xi
e^{\frac{2\pi^{2}l(4\mu+\nu)}{\beta}}+2e^{\frac{4\pi^{2}l\mu}{\beta}}-e^{\frac{8\pi^{2}l\mu}{\beta}}-1\right)}{1-e^{-\frac{4\pi^{2}l\nu}{\beta}}}}\right)\right),$
(4.5) $\displaystyle b_{c}(2)$
$\displaystyle\to\frac{\beta}{4\pi}\left(\log(2)-4i\pi
c_{1}-\log\left(\frac{e^{-\frac{4\pi^{2}l}{\beta}}\left(\xi
e^{\frac{2\pi^{2}l(4\mu+\nu)}{\beta}}+2e^{\frac{4\pi^{2}l\mu}{\beta}}-e^{\frac{8\pi^{2}l\mu}{\beta}}-1\right)}{1-e^{-\frac{4\pi^{2}l\nu}{\beta}}}\right)\right),$
(4.6) $\displaystyle b_{c}(3)$
$\displaystyle\to\frac{\beta}{4\pi}\left(\log(2)-4i\pi
c_{1}-2\log\left(-\sqrt{\frac{e^{-\frac{4\pi^{2}l}{\beta}}\left(\xi\left(-e^{\frac{2\pi^{2}l(4\mu+\nu)}{\beta}}\right)+2e^{\frac{4\pi^{2}l\mu}{\beta}}-e^{\frac{8\pi^{2}l\mu}{\beta}}-1\right)}{1-e^{-\frac{4\pi^{2}l\nu}{\beta}}}}\right)\right),$
(4.7) $\displaystyle b_{c}(4)$
$\displaystyle\to\frac{\beta}{4\pi}\left(\log(2)-4i\pi
c_{1}-\log\left(\frac{e^{-\frac{4\pi^{2}l}{\beta}}\left(\xi\left(-e^{\frac{2\pi^{2}l(4\mu+\nu)}{\beta}}\right)+2e^{\frac{4\pi^{2}l\mu}{\beta}}-e^{\frac{8\pi^{2}l\mu}{\beta}}-1\right)}{1-e^{-\frac{4\pi^{2}l\nu}{\beta}}}\right)\right),$
(4.8)

where in the above relation $\xi$ is

$\displaystyle\xi$
$\displaystyle=\sqrt{e^{-\frac{8\pi^{2}l(2\mu+\nu)}{\beta}}\left(-4e^{\frac{4\pi^{2}l(\mu+\nu)}{\beta}}+4e^{\frac{8\pi^{2}l(\mu+\nu)}{\beta}}-2e^{\frac{4\pi^{2}l(2\mu+\nu)}{\beta}}-4e^{\frac{4\pi^{2}l(3\mu+\nu)}{\beta}}+e^{\frac{4\pi^{2}l(4\mu+\nu)}{\beta}}+4e^{\frac{8\pi^{2}l\mu}{\beta}}+e^{\frac{4\pi^{2}l\nu}{\beta}}\right)}.$
(4.9)

The real part of these four solutions are shown in figure 5. It can be seen
that their phase space become smoother from $b_{c}(1)$ to $b_{c}(4)$.

Figure 5: The plots of solution of $b_{c}$ before the Page time. From left to
right, they are the plot of $b_{c}(1)$, $b_{c}(2)$, $b_{c}(3)$ and $b_{c}(4)$.
They show the behavior of the critical bath size $b_{c}$, in terms of the
sizes of the two mixed system, i.e, $\mu$ and $\nu$. Here $\beta$ and $l$ are
set to one.

If we replace $\beta$ in the above relations 4.5 with $\beta=\pi/\sqrt{\kappa
E}$ which then leads to the temperature $T_{H}=\frac{1}{\pi}\sqrt{\frac{8\pi
GE}{2\Phi_{r}}}$ [1], we get the plots shown in figure 6 which show the
behavior of $b_{c}$ versus $E$ in the four regimes.

Figure 6: The plots of solution of $b_{c}$ before the Page time versus energy,
$E$. Here we set $\mu=\nu=3$, $l=1$ and $\kappa=2$.

If we insist on getting positive $b_{c}$s, we need to tune the parameters,
like increasing $\kappa$ and decreasing the intervals $\mu$ and $\nu$, which
then the results are shown in figure 7, where for positive $E$ we get an
almost constant value for $b_{c}$ but the initial behavior for small values of
$E$ looks different.

Figure 7: The plots of solution of $b_{c}$ before the Page time versus energy,
$E$. Here we set $\mu=0.1$, $\nu=0.2$, $l=1$ and $\kappa=40$.

For the case of $\mu=\nu$ and by tuning $\kappa$ we can get another specific
behavior which is shown in figure 8.

Figure 8: The plots of solution of $b_{c}$ before the Page time versus energy,
$E$. Here we set $\mu=\nu=1$, $l=1$ and $\kappa=3$.

As expected from the Page curve and the results coming from negativity found
in [14], for cases where the regions of $\mu$ and $\nu$ are very big, and also
their ratio $\mu/\nu$ is big as in figure 9, the behavior would have a
linearly decreasing function. However, when their sizes or the ratio between
them is very small, the behavior is different. This difference is due to the
fact that the mutual information relative to negativity, overestimates the
contribution of classical correlations, which specifically for smaller size
would have more significant effects. For bigger sizes, the effect of this
difference would be negligible.

Figure 9: The behavior of $b_{c}$ before the Page time versus the $\mu$, while
$\nu=2\mu$.

Figure 10: The behavior of $b_{c}$ before the Page time, versus $\mu$, for
very small size of $\nu=0.0000001$, in the left, and for the case where
$\nu=0.0000001\mu$, in the right.

If we choose a small size for the subsystem angular size, $\nu$, then as shown
in figure 10, the critical size of the bath $b_{c}$ would decrease by
increasing the size of the system $\mu$. By increasing the size of $\nu$, then
the behavior of dropping of the critical bath size $b_{c}$ versus $\mu$, would
change to a linear decreasing function.

In general, from these plots, one could find that before the Page time, in
order to get a vanishing mutual information and to get the disconnected
phases, the critical size of the bath should be much bigger than the two
system size $\mu$ and $\nu$, while as we see from figure 17, this would not be
true for the case after the Page time.

So the case before the Page time corresponds to the case where $L_{B}\gg
L_{A}$ in [14], and there is no replica wormhole present in the gravitational
interpretation of the random matrix theory [18]. The pattern of the dominant
saddles could then be found by understanding the corresponding diagrams of
[14] in each phase. For the case before the Page time, the dominant diagram
for $\langle\text{Tr}\rho^{n}\rangle=L_{A}^{1-n}\propto\mu^{1-n}$ is shown in
equation 4.10 of [14]. This case corresponds to the positive partial transpose
(PPT) state with a vanishing logarithmic negativity (LN) as shown in the upper
region of the phase diagram, region $I$ of figure. 2 in [14], which has a
semicircular distribution.

Again it worths to mention here that as observed in [19], the mass of graviton
which play a significant role in the model of Hawking radiation would depend
on the angle between the island and the boundary. This angle is the parameter
that the partial reduction is being performed over in [1] which determines the
phase and the Page curve.

In the model of [19], where its metric can been written as

$\displaystyle
ds_{10}^{2}=L_{4}^{2}ds^{2}_{\text{AdS}_{4}}+f_{1}^{2}ds^{2}_{S_{1}}+f_{2}^{2}ds^{2}_{S_{2}}+4\rho^{2}dzd\bar{z},$
(4.10)

where the warp factors are

$\displaystyle L_{4}^{8}=16\frac{N_{1}N_{2}}{W^{2}},\ \ \ \
\rho^{8}=\frac{N_{1}N_{2}W^{2}}{h^{4}\hat{h}^{4}},\ \ \
f_{1}^{8}=16h^{8}\frac{N_{2}W^{2}}{N_{1}^{3}},\ \ \ \
f_{2}^{8}=16\hat{h}^{8}\frac{N_{1}W^{2}}{N_{2}^{3}},$ (4.11)

and

$\displaystyle W=\partial
h\bar{\partial}\hat{h}+\bar{\partial}h\partial\hat{h}=\partial\bar{\partial}(h\hat{h}),$
$\displaystyle N_{1}=2h\hat{h}|\partial h|^{2}-h^{2}W,$ $\displaystyle
N_{2}=2h\hat{h}|\partial\hat{h}|^{2}-\hat{h}^{2}W,$ (4.12)

and $h$ are related to $\kappa$ as

$\displaystyle
h=-\frac{1}{2}e^{z}\kappa-N\log\text{th}\left(\frac{i\pi}{4}-\frac{z}{2}\right)+c.c.$
$\displaystyle\hat{h}=\frac{1}{2}e^{z}\hat{\kappa}-N\log\text{th}\left(\frac{z}{2}\right)+c.c,$
(4.13)

the four phases can be analyzed as well.

There, a new quantity has been defined named $\alpha$ as the ratio of
$\alpha=N/K$, where $N$ is the number of D5 branes and also NS5 branes and
there are $2NK$ semi-infinite D3 branes where also $K$ is a parameter in the
function $h$ and $\hat{h}$ in the metric. These parameters then can lead to
the similar phase structures we observed in the previous section.

The dilaton in fact alway affects the surfaces of the island greatly. Also, in
the Karch-Randall brane setup, the mass of graviton $m_{g}^{2}$ would depend
on the angle between the extremal surface, where the island is located inside
the bulk and the boundary of $\text{AdS}_{5}$, i.e,
$\partial(\text{AdS}_{5})$. As we saw here, when the angle increases, the
critical distance between the two intervals $b_{c}$ should decrease and this
is compatible with the result of [20], as there also when the mass of graviton
increased the critical distance between the two subregion $D_{c}$ would
decrease. This is because the mass of graviton breaks diffeomorphism and is a
parameter for dissipation in the system, which as it increases, the
correlations diminish more and more, and therefore for keeping the mutual
information between the two regions strong enough to be non-zero, the distance
between them should decrease.

Also, in [19], it has been shown that there is a critical
$\alpha_{\text{crit}}$ and a critical angle $\nu$ and therefore a critical
mass for graviton $m_{\text{g,crit}}$, where below it, islands cannot exist
and this is compatible with the results in our next section, where $b_{c}$ can
get saturated in our diagrams there.

In [19], where the problem of island in higher dimension type IIB string
theory has also been investigated, the connections between the properties of
the island and mass of graviton has also been explored. In their setup, there
are two kinds of extremal surface which are either the Hartman-Maldacena
($S_{HM}$) or the quantum extremal island surface ($S_{I}$). They found that
in the setup of type IIB, when the dilaton varies, the mass of graviton then
cannot become arbitrary light. One subtlety worths to mention here is that
only the funnel form of black holes can be embedded into $5d$ bulk space while
“droplet”-like solutions cannot be encoded in the $1d$ higher case, which then
the contribution for each case in the several models we considered here could
be further tracked.

###  4.2 After the Page time

For the case after the Page time, from the relation for the mutual information
4.3 and entanglement entropies 4.1 and 4.2, we can find

$\displaystyle\sinh^{2}\frac{\pi}{\beta}\Big{(}2\pi\ell\mu+2b_{c}\Big{)}=\sinh\frac{\pi}{\beta}\Big{(}2\pi\ell(2\mu+\nu)+2b_{c}\Big{)}\sinh\frac{\pi}{\beta}\Big{(}2\pi\ell\nu+2b_{c}\Big{)},$
(4.14)

which again at these specific size of the bath, $b_{c}$, the mutual
information between the two intervals would vanish. The four solutions of the
above equation are

$\displaystyle b_{c}(1)$ $\displaystyle\to\frac{\beta}{2\pi}\left(2i\pi
c_{1}+\log\left(-\sqrt{\frac{e^{-\frac{4\pi^{2}l(2\mu+\nu)}{\beta}}\left(\left(e^{\frac{4\pi^{2}l\mu}{\beta}}-1\right)^{2}e^{\frac{4\pi^{2}l\nu}{\beta}}-\eta\right)}{2\left(e^{\frac{4\pi^{2}l\nu}{\beta}}-1\right)}}\right)\right),$
(4.15) $\displaystyle b_{c}(2)$ $\displaystyle\to\frac{\beta}{4\pi}\left(4i\pi
c_{1}+\log\left(\frac{e^{-\frac{4\pi^{2}l(2\mu+\nu)}{\beta}}\left(\left(e^{\frac{4\pi^{2}l\mu}{\beta}}-1\right)^{2}e^{\frac{4\pi^{2}l\nu}{\beta}}-\eta\right)}{2\left(e^{\frac{4\pi^{2}l\nu}{\beta}}-1\right)}\right)\right),$
(4.16) $\displaystyle b_{c}(3)$ $\displaystyle\to\frac{\beta}{2\pi}\left(2i\pi
c_{1}+\log\left(-\sqrt{\frac{e^{-\frac{4\pi^{2}l(2\mu+\nu)}{\beta}}\left(\left(e^{\frac{4\pi^{2}l\mu}{\beta}}-1\right)^{2}e^{\frac{4\pi^{2}l\nu}{\beta}}+\eta\right)}{2\left(e^{\frac{4\pi^{2}l\nu}{\beta}}-1\right)}}\right)\right),$
(4.17) $\displaystyle b_{c}(4)$ $\displaystyle\to\frac{\beta}{4\pi}\left(4i\pi
c_{1}+\log\left(\frac{e^{-\frac{4\pi^{2}l(2\mu+\nu)}{\beta}}\left(\left(e^{\frac{4\pi^{2}l\mu}{\beta}}-1\right)^{2}e^{\frac{4\pi^{2}l\nu}{\beta}}+\eta\right)}{2\left(e^{\frac{4\pi^{2}l\nu}{\beta}}-1\right)}\right)\right),$
(4.18)

where in the above relation $\eta$ is

$\displaystyle\eta$
$\displaystyle=\sqrt{-2e^{\frac{8\pi^{2}l(\mu+\nu)}{\beta}}+4e^{\frac{4\pi^{2}l(2\mu+\nu)}{\beta}}+e^{\frac{8\pi^{2}l(2\mu+\nu)}{\beta}}-4e^{\frac{4\pi^{2}l(\mu+2\nu)}{\beta}}-4e^{\frac{4\pi^{2}l(3\mu+2\nu)}{\beta}}+4e^{\frac{4\pi^{2}l(2\mu+3\nu)}{\beta}}+e^{\frac{8\pi^{2}l\nu}{\beta}}}.$
(4.19)

Figure 11: The plots of solution of $b_{c}$ after the Page time are shown.
From left to right are the plots of $b_{c}(1)$, $b_{c}(2)$, $b_{c}(3)$ and
$b_{c}(4)$. These plots show the behavior of the critical bath size $b_{c}$,
in terms of the two mixed system size $\mu$ and $\nu$. Here $\beta$ and $l$
are set to one.

The real part of these four solutions are shown in figure 11. Similar to the
previous case, the phase space becomes smoother from the case of $b_{c}(1)$ to
$b_{c}(4)$. If one fixes the ratio of the two system sizes as $\nu/\mu=0.001$,
the behavior of $b_{c}$ would be as sown in the right side of figure 11.

Inserting $\beta=\pi/\sqrt{\kappa E}$ in the above relations 4.15, 4.16, 4.17
and 4.19 with $\beta=\pi/\sqrt{\kappa E}$, we get the plots shown in figure 12
which show the behavior of $b_{c}$ versus $E$ in the four regimes. One could
see that for the first two solutions of $b_{c_{1}}$ and $b_{c_{2}}$, the
behavior of $b_{c}$ versus $E$ are critically different.

Figure 12: The plots of $b_{c}$ versus energy $E$ for the cases after the Page
time, for $\mu=\nu=3$, $l=1$ and $\kappa=2$.

For the case after the Page time we could not tune energy to get positive
values of $b_{c}$ but the results are shown in figure 13. One should note that
for negative energies the results would not be physical.

Figure 13: The plots of $b_{c}$ versus energy $E$ for the cases after the Page
time, for $\mu=0.1$, $\nu=0.2$, $l=1$ and $\kappa=40$.

For the case of $\mu=\nu=1$ and $\kappa=3$ we can get a third behavior shown
in figure 14.

Figure 14: The plots of $b_{c}$ versus energy, $E$, for the cases after the
Page time, for $\mu=\nu=1$, $l=1$ and $\kappa=3$. Figure 15: The behavior of
$b_{c}$ after the Page time, versus $\mu$, for $\nu=0.1\mu$. This behavior is
very similar to phase diagram found using negativity in [14].

Figure 16: The behavior of $b_{c}$ after the Page time, versus $\mu$, for very
small size of $\nu=0.01$, in the left and for the ratio $\nu=0.001\mu$, in the
right.

The behavior of the critical size of the bath, $b_{c}$, versus the system size
$\mu$, for the small size of the other system $\nu=0.01$ is shown in figure
17. It can be seen that it would increase at the beginning and then, as shown
in figure 10, it becomes constant for the larger values of $\mu$, which is
different from the case before the Page time.

Similar to [14], a dimensionless tuning parameter $q=b_{c}/\mu$ could be
defined which would probe the phase space in different setups.

It also worths to mention that for the cases that $b_{c}$ is positive,
increasing $\beta$ would increase $b_{c}$. This is because when the
temperature decreases, as also noticed in [20], the correlation could sustain
more, and therefore the critical distance between intervals could increase.

Note that the first solution, $b_{c_{1}}$ is specially related to the kink
mirror solution [16, 21], where in addition to considerations of the
symmetries, entanglement structure and the form of energy momentum tensor, the
behavior of $b_{c_{1}}$ versus $\beta$ shows the same behavior.

In principle, in the classifications of the regimes, the factor $n$ in the
mapping functions of [16], would translate to the late time behavior of
intervals $\mu$ and $\nu$ in our studies, as we discussed various scenarios
above.

Figure 17: The behavior of $b_{c_{1}}$ versus $\beta$ in the left, and $p(u)$
versus $\beta$, in the right, [16, 21].

From figure 17, one could also see that for the case after the Page time, the
critical bath size, compared to $\mu$ and $\nu$, would be very small or more
precisely, for getting a vanishing mutual information and the disconnected
phases, the tuning parameter $q$ would be much smaller. This is due to the
fact that after Page time most of the degrees of freedom of black hole are
already evaporated and therefore a smaller size of bath would be enough for
thermalizing the system and also there would be less quantum correlations
between $\mu$ and $\nu$ in this case.

For this scenario of states after the Page time which corresponds to the case
where $L_{A}\gg L_{B}$ in [14], the replica wormhole would be present. Also,
the dominant diagram for
$\langle\text{Tr}\rho^{n}\rangle=L_{B}^{1-n}\propto{b_{c}}^{1-n}$, would be
the one shown in equation 4.9 of [14]. This case corresponds to the negative
partial transpose (NPT) state with a non-vanishing logarithmic negativity (LN)
as shown in the lower regions of the phase diagram, i.e, the regions $II$ and
$III$ in figure. 2 of [4], which depending on the size of $\mu$ or $\nu$ would
have different spectrum distributions, namely maximally entangled and
entanglement saturtaion cases. In the first case when $\mu$ is small or $\nu$
is very large, we have two disjoint Marcenko-Pastur-like distributions, and
when $\mu$ is large or $\nu$ is very small, there would be the semicircular
distribution with partly negative domain, corresponding to the saturated
entanglement. From our studies here though, these two cases cannot be
distinguished.

In [22], the critical size of bath, where a phase transition occurs has also
been discussed but in a different context, where there is a brane and the
corresponding defect for it. There, the critical size of the bath depends on
the tension of the brane while here it depends on the relative size of the two
subsystems $\mu$ and $\nu$ and also their sizes compared to the bath. There,
also it has been confirmed that the Page time would depend on the ratio of the
central charges of the defect brane versus the central charge of the system,
which agrees with our results here.

In [23], the complexity for a single-sided AdS black hole with an EOW brane
has been calculated where the late time behavior of complexity is controlled
by the parameter $e^{S_{0}}$. This parameter would completely depend on the
tension of the EOW brane sitting at the end of the geometry with JT action. In
our relations 4.5 and also 4.15, 4.16, 4.17, 4.19, this term affects through
$\beta$, as it follows the relation $\beta=\pi/\sqrt{\kappa E}$, where $E=M$
is the mass of the black hole and $\kappa$ is a parameter related to the
tension of the brane.

For our setup with such angles, for each interval we can write

$\displaystyle\langle\mu(t)\rangle=-\underset{\Delta\to
0}{\text{lim}}\frac{\partial\langle\varsigma(t)\varsigma(0)\rangle_{\text{non-
int}}}{\partial\Delta},$ (4.20)

where $\Delta$ is the scaling dimension for the operator $\varsigma$. So for
two systems, with angular interval sizes of $\mu$ and $\nu$, the mixed
correlation structure and critical size $b_{c}$, and also the time needed for
the saturation in a dynamical setup, would all depend on $e^{S_{0}}$ and the
tension of the brane.

Finally, it worths to mention here the four saddles have even been observed in
new models of complexity constructed in [24], where the behavior of Krylov
operator complexity or complexity-spreading and Krylov entanglement, or
entanglement spreading (which they could be connected to each other through
the exponential behavior), both show the same behaviors, having four regimes
of ramp, peak, slope and plateau.

###  4.3 Comparing with results from partial transpose

The studies of [14] has been done using partial transpose (PT) and logarithmic
negativity (LN) and here we use mutual information and critical size of bath.
By comparing our plots, one could deduce further results from various
entanglement phase transitions which worths to mention here.

Note that in general the mutual information relative to PT and LN would
overestimate the entanglement of classically correlated states which are
separable, relative to the case of LN and PT. The separable state which is a
completely classical state could be written as

$\displaystyle\rho_{\text{sep}}=\sum_{i,j}p_{i,j}\rho_{1}^{(i)}\otimes\rho_{2}^{(j)},\
\ \ \ \ \ \ p_{ij}>0.$ (4.21)

One then can use the random induced mixed states setup of [14] where the
random induced mixed states $\\{\rho_{A}\\}$ in the Hilbert space of
$\mathcal{H}_{A}=\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}$ can be found
by partial tracing of the whole pure states in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Then, in a tensor product basis, a
random pure state could be written as

$\displaystyle\ket{\Psi}=\sum_{i=1}^{L_{A}}\sum_{\alpha=1}^{L_{B}}X_{i\alpha}\ket{\Psi_{A}^{(i)}}\otimes\ket{\Psi_{B}^{(\alpha)}},$
(4.22)

where $X$ are $L_{A}\times L_{B}$ rectangular random matrix with elements
$X_{i\alpha}$ which are independent Gaussian random complex variables with the
joint probability density as

$\displaystyle
P(\\{X_{i\alpha}\\})=\mathcal{Z}^{-1}\exp\\{-L_{A}L_{B}\text{Tr}(XX^{\dagger})\\},$
(4.23)

where here $L_{A}=L_{\mu}\times L_{\nu}$ in our case, and the density matrix
$\rho_{A}$ could be written as
$\rho_{A}=\frac{XX^{\dagger}}{\text{Tr}(XX^{\dagger})}$.

Then, the spectral density of eigenvalues $\\{\lambda_{i}\\}$ would follow the
relation

$\displaystyle
P(\lambda)=\sum_{i=1}^{L_{A}}\delta(\lambda-\lambda_{i})=\frac{qL_{A}^{2}}{2\pi}\frac{\sqrt{(\lambda_{+}-\lambda)(\lambda-\lambda_{-})}}{\lambda},$
$\displaystyle\ \lambda_{\pm}=\frac{1}{L_{A}}(1\pm 1/\sqrt{q})^{2},\ \ \
\lambda\in[\lambda_{-},\lambda_{+}],\ \ \ \ q=L_{B}/L_{A}\geq 1,$ (4.24)

which this behavior is compatible with the behavior of critical bath size
$b_{c}$ in our setup. Ssimilarly for the case of $q<1$, the entanglement
spectrum would have a delta-function at the origin which is exactly what we
got here in the diagrams of $b_{c}$, for the case after the Page time where
$q$ becomes much smaller than one.

###  4.4 Quantum error corrections of black hole from $1d$ higher point of
view

In [25], for the JT gravity model, the quantum error correction properties of
the black hole interior which is entangled to a non-gravitational bath have
been studied and it was shown that the interior is robust against the generic
and low-rank operations.

They argued that after the Page time, the information of the interior degrees
of freedom of the black hole and those in the bath are connected, and as the
information of the black hole get encoded in the bath Hilbert space, so the
noise in the density matrix of the interior of black hole coming from quantum
operations on the bath can be corrected. The bound on the noise would depend
on the black hole entropy and the code subspace dimension.

So the holographic quantum channels in this case should satisfy the “Knill-
Laflamme” conditions

$\displaystyle
P_{\text{code}}E_{m}^{\dagger}E_{n}P_{\text{code}}=\alpha_{mn}P_{\text{code}},$
(4.25)

where $P_{\text{code}}=VV^{\dagger}$ is the projector onto the code subspace.

Since there is a phase transition in the behavior of mutual information and
the saddles of Euclidean path integrals, the factorization and decoupling
behavior of the total Hilbert space of the black hole and therefore the
behavior of the holographic quantum error correction channels and codes would
change for the case before and after the Page time. Indeed, after the Page
time the correctability of errors would significantly change.

For the JT gravity in [25], it was also found that the recovery channel
$\mathcal{R}$ would satisfy the relation

$\displaystyle\text{max}_{\rho}||\mathcal{R}(\mathcal{E}(\rho))-\rho||_{1}\leq\epsilon,\
\ \ \ \ \epsilon\sim e^{-S_{0}/4},$ (4.26)

so the bigger the entropy of the higher-dimensional extremal black hole (where
the JT theory would be the result of its two-dimensional reduction), the lower
the errors of the holographic quantum recovery channels would be and so the
more robust the black hole would be against the noises in the bath. The
parameter $S_{0}$ here could be considered as the ground state entropy of the
JT system. This relation then led to $2(t-1)\leq(\mathcal{S}-\log_{2}d)$ [25],
where $t-1$ is the maximum number of qubits in the physical Hilbert space
which can tolerate the error, $d$ is the dimension of the code space and
$2^{\mathcal{S}}$ is the dimension of the physical Hilbert space.

The parameter $t$ also determines how much of the information of the $3d$ BTZ
black hole would actually penetrate to the lower dimensional $2d$ JT gravity
case, which we will investigate this point further in the next section. This
penetration of information between various dimensions can also be understood
by studying the partition functions, the logarithmic negativity, and also
similar to [25], the Rényi entropy, i.e,
$S^{(n)}=\frac{1}{1-n}\log\text{Tr}\rho^{n}$ from the Rényi mutual
information.

If similar to [25], we write the reduced density matrix on the interior and
exterior code subspace, and also on the environmental part, with the
normalization constant $\mathcal{N}$, one would get the relation
$\mathcal{N}=d_{i}d_{e}e^{S_{0}}Z_{1}$, where $d_{i}$ and $d_{e}$ are the
dimensions of the interior and exterior factors in the code subspace and
$Z_{1}$ is the exponential of the on-shell, JT gravitational action on a disc
which is capped off by an EOW brane, and some additional non-code-subspace
bulk field theory modes in the Hartle-Hawking vacuum state. So, one could
consider the information of the $2d$ case gathered in the $Z_{1}$ part, times
the factor of $d_{i}d_{e}e^{S_{0}}$ leading to the higher $3d$ case where the
non-robust information are gathered in the non-code subspace section.

Then, according to the result of [25] and based on the discussions on erasures
of errors far from the phase transition points, the $\mathbb{Z}_{n}$-breaking
bulk geometries cannot dominate. The two dominant geometries out of the four
are the fully disconnected and the fully connected ones. Both of them preserve
the $\mathbb{Z}_{n}$ symmetry of the asymptotic boundary conditions. The
preservation of this symmetry is the reason that from the $3d$ BTZ point of
view in [1], the phase structure of $2d$ JT gravity can be obtained.

So, approximately, the reduced density matrix of the $3d$ case can be written
as

$\displaystyle\text{Tr}\rho^{\prime
n}_{\mathfrak{R}}=\frac{1}{k^{n}\mathcal{N}^{n}}\sum_{i_{1},...,i_{n}}\sum_{i^{\prime}_{1},...,i^{\prime}_{n}}\sum_{\alpha_{1},...,\alpha_{n}}\langle\psi^{\alpha_{1}}_{i_{2},i^{\prime}_{1}}|\psi^{\alpha_{1}}_{i_{1},i^{\prime}_{1}}\rangle_{B}...\langle\psi^{\alpha_{n}}_{i_{1},i^{\prime}_{n}}|\psi^{\alpha_{n}}_{i_{n},i^{\prime}_{n}}\rangle_{B},$
(4.27)

which could be considered as an ensemble average of the microstates of the
$2d$ JT gravity plus the EOW branes. When the parameters $k$ and $S_{0}$
become larger, more and more information of the $3d$ geometry is encoded in
the $2d$ JT section.

Based on the quantum error correction idea of [25], the connected contribution
of the partition function would dominate the disconnected one when

$\displaystyle\frac{(\frac{\ell}{k})^{n-1}+k^{n-1}}{1+\ell^{n-1}}\approx\left(\frac{k}{\ell}\right)^{n-1}\ll
e^{(1-n)(S_{BH}+\log d_{i})},$ (4.28)

where $d_{i}$ and $d_{e}$ are the dimensions of the interior and exterior
factors in the code subspace, and $n$ is the order of the Renyi entropy.

Here the Renyi entropy is defined over the random variables of the set
$\mathcal{A}=\\{x_{1},x_{2},...,x_{n}\\}$ where $S_{n}$ is a non-increasing
function of $n$. Also, $\ell$ is the dimension of the environmental factor and
$k$ controls the dimension of the radiation subspace that is entangled with
the black hole microstates.

This relation then could be extended for the four phase scenarios of the mixed
systems, and therefore the quantum error correction relations could be applied
there, as we try to explain next.

So using the results of [6], for the case of transitioning from $g=I$ to
$g=\tau$, for the even values of $m$, we should have the inequality

$\displaystyle\frac{Z_{2}^{m}}{Z_{1}^{2m}}\sim
e^{(1-m)S_{BH}}>\frac{d_{e}^{m-1}}{k^{m}C_{m}},$ (4.29)

and for the odd case the condition would be

$\displaystyle\frac{Z_{2}^{m-1}}{Z_{1}^{2m-2}}>\frac{d_{e}^{m-1}}{k^{m-1}C_{m-1}(2m-1)}.$
(4.30)

Note that here $k=k_{1}k_{2}$, and also $C_{m}=\frac{1}{m+1}\binom{2m}{m}$ is
the Catalan number which is the number of non-crossing pairings here. For the
disconnected and pairwise connected case, $d_{e}^{1-n}$ controls the behavior
of the partition functions.

For going from $g=X$ to $g=\tau$, for the odd case then we should have the
relation

$\displaystyle\frac{Z_{2m-1}}{Z_{1}Z_{2}^{m-1}}<\frac{(2m-1)C_{m-1}}{d_{i}^{m}d_{e}^{m}k^{m-1}},$
(4.31)

while for the even case, the inequality

$\displaystyle\frac{Z_{2m}}{Z_{2}^{m}}<\frac{k_{2}^{2m-2}C_{m}}{d_{i}^{m}d_{e}^{m}k^{m-1}},$
(4.32)

should be satisfied, which unlike the odd case is independent of the $Z_{1}$.

For going from $g=X^{-1}$ to $g=\tau$, one only needs to swap $k_{2}$ to
$k_{1}$ in the above relations. Note that $g=I$ cannot reach to $g=X$ or
$g=X^{-1}$, and also $g=X$ is further away from $g=X^{-1}$, which can also be
seen from the relations between the dimensions of the code subspaces.

Therefore, for a mixed setup, the lower bounds for each saddle, would depend
in a rather complicated way to the number of microstates in each case, i.e,
$k_{i}$, and the dimension of the code subspace of the interior or exterior,
depending whether we are in the connected or the disconnected phase.

##  5 Comparing partition functions and density states of $3d$ versus $2d$

In [18], for the computation of the purity, using the gravity amplitudes in
$2d$ JT gravity, the density matrix and also the Rényi 2-entropy have been
calculated where it was argued that for the $2d$ case, there are two ways of
filling the boundary condition. In this section we would like to compare this
procedure for the case of $3d$ gravity and specifically for the
$\text{AdS}_{3}$ and BTZ solutions and then compare the result with the case
of dimensional reduction from BTZ to $2d$ JT gravity of [1], in order to see
how much of the information, phase structures, and patterns of entanglement
would project from the higher dimensions to the lower dimensions, and to
connect the different parameters of these theories.

Note that the theories in $3d$ and $2d$ have been connected together through
either holography or dimensional reduction, where these connections have been
extensively studied in [26]. The $3d$ gravity would lead to $2d$ JT gravity by
dimensional reduction and is connected to $2d$ Liouville theory by holography.
The $2d$ Liouville CFT would then lead to $1d$ Schwarzian theory by
dimensional reduction.

One the other hand the $3d$ Chern-Simons theory would be related to $2d$ BF
theory by dimensional reduction and to $2d$ WZW model by holography. The $2d$
WZW model after the dimensional reduction would lead to the model for $1d$
particle on group.

An example of $1d$ case is also the BFSS matrix model which arises in nearly
$\text{AdS}_{2}/\text{CFT}_{1}$ models. Note that one can even go to lower
dimensions, down to $D(-1)$-branes in type IIB string theory knows as IKKT
matrix model. In [27], also the connections between $2d$ JT gravity and the
$c<1$ limit of $2d$ string theory where its world-sheet theory consists of a
spacelike Liouville CFT, coupled to a non-rational time-like Liouville CFT,
has been studied. Also, the $1d$ Schwarzian model can be connected to $1d$
random matrix model such as BFSS or $1d$ SYK model.

The dynamics governed completely by the Schwarzian derivative action

$\displaystyle S=-C\int d\tau\\{F,\tau\\},$ (5.1)

where $C=\frac{a}{16\pi G}$, and
$\\{F,\tau\\}=\frac{F^{\prime\prime\prime}}{F^{\prime}}-\frac{3}{2}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2}$,
where $F(\tau)$ is the time reparametrization which then becomes the physical
and dynamical degree of freedom, and $a$ is a constant [28]. This relation is
completely identical to the Chern-Simons/WZW topological duality where the
large gauge transformation of Chern-Simons theory becomes the dynamical
degrees of freedom of WZW model. Now the goal is to understand the
interconnections between these dynamics, the island formation and black hole
evolutions in each case. One way to see these connections is to compute the
boundary correlators of the thermal JT theory [28]

$\displaystyle\langle\mathcal{O}_{h_{1}}\mathcal{O}_{h_{2}}...\rangle_{\beta}=\frac{1}{Z}\int_{\mathcal{M}}[\mathcal{D}F]\mathcal{O}_{h_{1}}\mathcal{O}_{h_{2}}...e^{C\int_{0}^{\beta}d\tau\\{F,\tau\\}}$
(5.2)

and compare with the corresponding ones in other theories.

When one writes the JT gravity in terms of the first order formulation of
$SL(2,\mathbb{R})$ BF formulation of JT gravity, the operators
$\mathcal{O}_{h_{i}}$ can be considered as the boundary anchored Wilson line.
The Wilson line of this operator can be written as

$\displaystyle\mathcal{O}_{h}(\tau_{1},\tau_{2})\equiv\left(\frac{F^{\prime}(\tau_{1})F^{\prime}(\tau_{2})}{(F(\tau_{1})-F(\tau_{2}))^{2}}\right)^{h},$
(5.3)

where $F(\tau)$ is time reparametrization for each end point of the Wilson
line. The value of $h$ for the case where both of these end points are on the
manifold $M$ (or $Q$), or one of them ends on $M$ and the other on $Q$, would
be different. Therefore, this boundary two-point function and its holographic
dual can probe the phase structures of black hole radiation and island
formation. The genus-zero partition function of JT gravity for each scenario
is

$\displaystyle\frac{1}{Z}\int dE_{2}e^{-\beta E_{2}}\rho_{0}(E_{2})\int
dE_{1}\rho_{0}(E_{1})e^{-\tau(E_{1}-E_{2})}\frac{\Gamma(h\pm i\sqrt{E_{1}}\pm
i\sqrt{E_{2}})}{\Gamma(2h)},\ \ \ \ \ \tau=\tau_{2}-\tau_{1},$ (5.4)

where $Z$ is the Schwarzian disk partition function, i.e, genus $0$, and
$\rho_{0}(E)=\frac{1}{2\pi^{2}}\sinh 2\pi\sqrt{E}$. However, for the mixed
systems, or for the case where the island appears (or it becomes outside of
the black hole horizon,) this relation is not complete. Specifically the order
of the weight of the operators $h_{i}$ for the case ending on manifold $M$
which is the physical subsystem would be completely different from the case
when the Wilson line ends on the manifold $Q$ where the island forms. In
addition, this shows that only the “massive” fields (graviton, scalar or
fermion, etc) can detect the island formation as also shown in [19], from
other methods. Therefore, the above relation for the mixed case, and also for
the evolutions after the Page time become more complicated. We propose that,
for the first order of approximation, this relation would become

$\frac{1}{Z_{M}}\frac{1}{Z_{Q}}\frac{1}{Z_{QM}}\times\\\
\int(dE_{2})_{M}e^{-\beta_{M}(E_{2})_{M}}\rho_{0}(E_{2})_{M}\int
d(E_{1})_{M}\rho_{0}((E_{1})_{M})e^{-\tau((E_{1})_{M}-(E_{2})_{M})}\frac{\Gamma(h_{M}\pm
i\sqrt{(E_{1})_{M}}\pm i\sqrt{(E_{2})_{M}})}{\Gamma(2h_{M})}\times\\\
\int(dE_{2})_{Q}e^{-\beta_{Q}(E_{2})_{Q}}\rho_{0}(E_{2})_{Q}\int
d(E_{1})_{Q}\rho_{0}((E_{1})_{Q})e^{-\tau((E_{1})_{Q}-(E_{2})_{Q})}\frac{\Gamma(h_{Q}\pm
i\sqrt{(E_{1})_{Q}}\pm i\sqrt{(E_{2})_{Q}})}{\Gamma(2h_{Q})}\times\\\
\int(dE_{2})_{QM}e^{-\beta_{QM}(E_{2})_{QM}}\rho_{0}(E_{2})_{QM}\times\\\ \int
d(E_{1})_{QM}\rho_{0}((E_{1})_{QM})e^{-\tau((E_{1})_{QM}-(E_{2})_{QM})}\frac{\Gamma(h_{QM}\pm
i\sqrt{(E_{1})_{QM}}\pm i\sqrt{(E_{2})_{QM}})}{\Gamma(2h_{QM})},$ (5.5)

where the label $M$ specifies the parameters of the operators which are
inserted on the manifold $M$, and $Q$ labels those which end on the end of the
world brane where the island is being created, as shown in figure 28.

Similar to [27], the connections between $2d$ WZW model and $2d$ BF theory can
then be constructed. In addition, The model of $1d$ particle on group could be
related to some matrix-like model.

So similar to the story depicted in [1] which embedded the JT gravity in $3d$
BTZ black hole geometry, the $1d$ Schwarzian model can be embedded in $2d$
Liouville theory and the corresponding flow of information could get tracked.
On the other hand the story of [1] can be connected to compact group
construction using the pictures in [26].

Figure 18: The connections between various theories in higher dimensions and
$1d$ lower dimension case, through holography or dimensional reduction. Inside
the torus is the $3d$ Chern-Simons gravity where its boundary is $2d$
Liouville gravity or $2d$ WZW theory in the case of topological theory. Also,
after the dimension reduction, the $3d$ gravity either leads to $2d$ JT
gravity or to the $2d$ BF theory. These two theories also have a $1d$ lower
dimension boundary theory, namely $1d$ Schwarzian, or $1d$ particle on group
theory.

Now, using the figure 18 which depicts the connections between various
theories, we can study the problem of dynamical evaporation of black holes
similar to [1] for the case of $3d$ Chern-Simons, $2d$ WZW and $1d$ particle
on group, as one of the motivations of this work is to investigate the island
formation, black hole evolution, and saddles of mixed states of black hole
radiation using gauge theory, WZW model and its connections to the JT gravity.

###  5.1 The $3d$ Chern-Simons gravity to $1d$ particle on group

In [1] the $1d$ Schwarzian action has been derived from three dimensional
gravity. We wish here to do the same for the Chern-Simons $3d$ gravity leading
to $1d$ particle on a group.

So, first we study the dimensional reduction from $3d$ to $2d$ WZW. The Chern-
Simons action is

$\displaystyle CS[A]=\frac{k}{4\pi}\int_{\mathcal{M}}\text{tr}\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge A\right).$ (5.6)

The total gauge-invariant action for this theory is

$\displaystyle\widetilde{CS}[A,g]=CS[A]+WZNW[g]+\frac{k}{4\pi}\int_{\partial\mathcal{M}}\text{tr}\left(2A_{z}\partial_{\bar{z}}gg^{-1}+A_{\bar{z}}A_{z}\right)d^{2}z,$
(5.7)

where the Wess-Zumino-Novikov-Witten (WZNW) action is

$\displaystyle
WZNW[g]=\frac{k}{4\pi}\int_{\partial\mathcal{M}}\text{tr}\left(\partial_{z}gg^{-1}\partial_{\bar{z}}gg^{-1}\right)d^{2}z-\frac{k}{12\pi}\int_{\mathcal{M}}\text{tr}\left(dgd^{-1}\right)^{3}.$
(5.8)

The simplest $2d$ gravity model would be de-Sitter, which is characterized by
$D=\Phi$, $V\propto\Phi$ and $Z\equiv 0$. In terms of the non-abelian gauge
theory as in BF-type, it could be written as

$\displaystyle L_{deS}[A,B]=-2\int_{\mathcal{M}}\text{tr}(BF),$ (5.9)

where $F=dA+A\wedge A$ is the curvature two-form of the standard gauge field
$A$, and $B$ is a function on the $2d$ manifold $\mathcal{M}$.

The boundary term which similar to the Gibbons-Hawking term would be necessary
to add to the above action to make it gauge invariant can be written in the
following form

$\displaystyle
S_{\text{boundary}}=2\int_{\partial\mathcal{M}}\text{tr}\left(B(A+dgg^{-1})\right).$
(5.10)

So the whole action would be

$\displaystyle\widetilde{L_{\text{deS}}}[A,B,g]=-2\int_{\mathcal{M}}\text{tr}(BF)+2\int_{\partial\mathcal{M}}\text{tr}\left(B(A+dgg^{-1})\right).$
(5.11)

This corresponds to the Gibbons-Hawking term, i.e, equation 2.24 of [1]. This
action is invariant under $g\to g^{h}=h^{-1}g$.

Also, one can write the $3d$ CS theory as

$\displaystyle
S_{CS}\sim\int_{M_{3}}d^{3}x\epsilon^{ijk}A_{i}\partial_{j}A_{k},$ (5.12)

which by setting $A_{\phi}\chi$ and $\partial_{\phi}$ to zero, one can get the
$2d$ BF theory as

$\displaystyle S=\int_{M}d^{2}x\chi F+\frac{1}{2}\oint_{\partial M}dt\chi
A_{0}.$ (5.13)

For the matter action in [26], the boundary term has been found as

$\displaystyle S_{\text{matter}}=-\oint_{\partial M}dt\sigma
J^{r}=-\oint_{\partial M}dt\ \sigma(J_{+}-J_{-}),$ (5.14)

which is the net inward flux of charge which corresponds to the relation 2.24
of [1],

$\displaystyle S_{GH}=\frac{2\pi\alpha\ell}{8\pi G^{(3)}}\int
dt\sqrt{-h_{tt}}\phi_{b}\left(K^{(3)}+\frac{2}{\ell}\right),$ (5.15)

representing the flux of dilaton through the curvature of the geometry.

Additionally, similar to the $3d$ to $1d$ case of [1] for the Einstein theory,
the non-abelian BF theory could be written as

$\displaystyle S=\int_{M}d^{2}x\text{Tr}\chi F+\frac{1}{2}\oint_{\partial
M}dt\text{Tr}\chi A_{0},$ (5.16)

which after applying the boundary condition reduces to a particle on a group
manifold action as

$\displaystyle S=\frac{1}{2}\oint_{\partial
M}dt\text{Tr}(g^{-1}\partial_{t}g)^{2}.$ (5.17)

The only difference that this boundary term can create would be in the
Hamiltonian part $H=H_{\text{grav}}+H_{CS}$ leading to just a shift in the
energy as

$\displaystyle T_{tt}=\frac{\dot{\sigma}^{2}}{2},$ (5.18)

which its effects on the mixed correlations, similar to the case of [1] can be
investigated. This injection or shift in the energy momentum tensor could be
considered as the quench in BCFTs and similar to [29], the entropy and energy
could be related by the first law of entanglement as

$\displaystyle T_{tt}(x_{A},t)=\underset{|x_{A}-x_{B}|\to 0}{\lim}\
\frac{3}{\pi|x_{A}-x_{B}|^{2}}\ .\ \Delta S_{A,B}(x_{A},x_{B},t).$ (5.19)

So additional entropy $\Delta
S=\frac{\pi}{6}\frac{\Delta\dot{\varphi}^{2}}{\underset{|d\varphi|\to
0}{\lim}\frac{1}{d\varphi^{2}}}$ would be created due to this injection of
energy, which for two strips and for the phase structure of the mixed system
would have a significant effect. This energy momentum tensor also affects the
flux to the bath and therefore the resulting linear and exponential
evaporation of the black hole, such as those studied in [1]

The story of the black hole evolution, Page curve and island formations could
also be studied through CS/WZW duality instead of $2d$ JT gravity model. The
WZW is more string theory based, where its symmetry algebra is an affine Lie
algebra, i.e, $G(z)\times G(\bar{z})$ symmetry. In string theory, the
$\text{SL}(2,\mathbb{R})/U(1)$ gauged WZW model have been interpreted in [30]
as the Witten’s two-dimensional Euclidean black hole. Geometrically, the WZ
term describes the torsion of the manifold, which then affects the phase
transitions and mixed correlations.

The black hole and the island both can be defined from the
$\text{SL}(2,\mathbb{R})$ group manifold [31] which could be written as the
hyperboloid as

$\displaystyle x_{-1}^{2}+x_{0}^{2}-x_{1}^{2}-x_{2}^{2}=1,$ (5.20)

where the black hole corresponds to the topological circle in the
$(x_{-1},x_{0})$ plane that never shrinks to zero. The island also should be
defined on the same plane. Both the BTZ black hole and the island would be
obtained by dividing the universal covering group by “two” $\mathbb{Z}$
orbifold actions. The group elements of $\text{SL}(2,\mathbb{R})$ would act as

$\displaystyle g\to
e^{\pi(r_{+}-r_{-})\sigma_{3}}ge^{\pi(r_{+}+r_{-})\sigma_{3}},$ (5.21)

where $\sigma_{3}$ is the diagonal Pauli matrices.

The symmetries in the WZW model could define the generators by exponentiating
$\sigma_{3}$ which then can lead to the energy and angular momentum
generators. This could also be written using the group element parametrization

$\displaystyle
g=e^{\frac{r_{+}-r_{-}}{2}(t+\phi)\sigma_{3}}e^{\rho\sigma_{1}}e^{-\frac{r_{+}+r_{-}}{2}(t-\phi)\sigma_{3}},$
(5.22)

where the radial coordinate $\rho$ is related to the BTZ radial coordinate $r$
through

$\displaystyle\cosh^{2}\rho=\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}.$
(5.23)

The geodesic length and therefore the entanglement entropy and mixed
correlation measures corresponding to the WZW case then can be defined using
these $g$ elements. On the other hand, similar to the usual case of the JT
gravity action coupled to a CFT bath, the corresponding “gauged” model
contains two BF systems coupled to each other. Then, the arising of the Page
curve in the group analysis and the orbifolded manifold could be found by
analyzing the geodesics.

Note that the JT gravity itself can be written as an equivalent Schwarzian
quantum mechanical theory on the holographic boundary. The JT gravity can be
considered as the s-wave dimensional reduction of the $3d$ pure gravity with
$\Lambda<0$.

###  5.2 The connections between JT model and Liouville gravity

First, we review the change of partition function during black hole evolution
and Page transition in $2d$ JT model as studied in [18], and then compare it
with the case of $2d$ BF theory and $2d$ WZW model.

The action in [18] was for a black hole in JT gravity with an EOW brane behind
the horizon which is a $\mathbb{Z}_{2}$ quotient of the two-sided black hole.
A particle which has the mass $\mu$, ($\mu\geq 0$) would be considered behind
the horizon. For this system, the Euclidean action can be written as

$\displaystyle I=I_{JT}+\mu\int_{\text{brane}}ds,$ (5.24)

where the JT action is

$\displaystyle
I_{JT}=-\frac{S_{0}}{2\pi}\left[\frac{1}{2}\int_{\mathcal{M}}\sqrt{g}R+\int_{\partial\mathcal{M}}\sqrt{h}K\right]-\left[\frac{1}{2}\int_{\mathcal{M}}\sqrt{g}\phi(R+2)+\int_{\partial\mathcal{M}}\sqrt{h}\phi
K\right].$ (5.25)

The partition function is

$\displaystyle
Z_{n}=e^{S_{0}}\int_{0}^{\infty}d\ell_{1}...d\ell_{2n}e^{\frac{\ell_{1}+...+\ell_{2n}}{2}}I_{2n}(\ell_{1},...,\ell_{2n})\varphi_{\beta}(\ell_{1})e^{-\mu\ell_{2}}...\varphi_{\beta}(\ell_{2n-1})e^{-\mu\ell_{2n}},$
(5.26)

where $I_{2n}$ are

$\displaystyle
I_{2n}(\ell_{1},...,\ell_{2n})=2^{2n}\int_{0}^{\infty}ds\rho(s)K_{2is}(4e^{-\frac{\ell_{1}}{2})}...K_{2is}(4e^{-\frac{\ell_{2n}}{2}}),\
\ \ \ \ \ \ \rho(s)=\frac{s}{2\pi^{2}}\sinh(2\pi s),$ (5.27)

and $\varphi_{\beta}(\ell)$ is

$\displaystyle\varphi_{\beta}(\ell)=4e^{-\frac{\ell}{2}}\int_{0}^{\infty}ds\rho(s)e^{-\frac{\beta
s^{2}}{2}}K_{2is}(4e^{-\frac{\ell}{2}}).$ (5.28)

Here, $K$ is the modified Bessel function and $\varphi_{\beta}$s are the
Hartle-Hawking state in the geodesic basis, and $I_{2n}$ consists of $n$
geodesics on the EOW branes and $n$ geodesics that should be glued to form the
Hartle-Hawking states.

If one assumes that the EOW brane has a very large number of internal and
orthogonal states $k$, the Page transition would be between $\log(k)$ and
$S_{BH}\sim S_{0}+2\pi\phi$. So the island formulation for the JT predicts

$\displaystyle S(R)=\text{min}\\{\log(k),S_{BH}\\}.$ (5.29)

So the transition is between the two and a function of $k$, where then also
after dimension reduction it would change, such as those examples in [32].

So the relations we can observe between these theories are dual to the
equivalence between the three definitions of the boundary entropy explained in
[33, 34], i.e, between the disk amplitude, $S_{\text{bdy}(\alpha)}=\log
g_{\alpha}$, where $g_{\alpha}\equiv\langle 0|B_{\alpha}\rangle$, and cylinder
amplitude $Z_{(\alpha,\beta)}^{\text{cylinder}}=\langle
B_{\alpha}|e^{-HL}|B_{\beta}\rangle\approx g_{\alpha}g_{\beta}e^{-E_{0}L}$,
where $g_{\alpha}g_{\beta}$ is related to the boundary part and $e^{-E_{0}L}$
is related to the bulk part, and finally to the usual entanglement entropy
relation of $2d$ CFT case, $S_{A}=\frac{c}{6}\log\frac{l}{\epsilon}+\log
g_{\alpha}$.

The level $k$ in the WZW model is actually related to this
$g_{\alpha}g_{\beta}$ of the boundary part, which then after the dimension
reduction further, would be related to the term of the $1d$ particle on a
group manifold, i.e, $\text{Tr}(g^{-1}\partial_{t}g)^{2}$.

In the $\text{AdS}_{3}/\text{BCFT}_{2}$ setup, the boundary entropy which is a
measure of degrees of freedom at the boundary can be written as
$g=e^{S_{\text{bdy}}}$ where $g$ is a function that is monotonically
decreasing under the boundary RG flow and is a function of the boundary
conditions labeled by $\alpha$. The movement of the island from inside of the
black hole toward the horizon will decrease the degrees of freedom on the
boundary brane, as it enforces this flow leading to this monotonic behavior.
Another interesting connection between the wormhole and island is that in the
replica trick calculation of the von Neumann entropy, by taking $n\to 1$ the
mouth of the wormhole would become the island itself, which again leads to the
result found before that by changing the topology and by increasing the genus,
the island can move inside of the horizon.

Then, the $2d$ Liouville gravity which can be interpreted as the quantum (q)
deformation of JT gravity, and also a specific model of the sinh dilaton
gravity [35] can be written as

$\displaystyle
S_{L}=\frac{1}{4\pi}\int_{\Sigma}\left[(\tilde{\nabla}\phi)^{2}+Q\tilde{R}\phi+4\pi\mu
e^{2b\phi}\right],\ \ \ Q=b+b^{-1},\ \ c_{L}=1+Q^{2},$ (5.30)

where $b$ is the parameter which defines the Liouville theory, and $\mu$ here
is the boundary cosmological constant. For building the FZZT-brane boundary
model, a boundary term as

$\displaystyle\frac{1}{2\pi}\oint_{\partial\Sigma}\left[Q\tilde{K}\phi+2\pi\mu
e^{b\phi}\right],$ (5.31)

is being added to the Liouville action 5.30, where $\mu_{B}$ is the boundary
cosomological constant. The metric and the field $b$ are related as
$ds^{2}e^{2b\phi{dzd\bar{z}}}$, and the boundary length is $\oint e^{b\phi}$,
and these parameters could be looked at from one-dimension higher and the
compactification point of view. For the Liouville theory, the entanglement
entropy with the presence of an island has been calculated in [36], where as
shown in figure 22, it can be seen that by increasing the parameters of the
potential, $\mu$ and $b$, the entanglement entropy of the island would
decrease, and since $b$ is in the exponent, it has a higher effect on breaking
the correlations. This Liouville potential $C\mu e^{b\phi}$ would reflect the
energy eigenvectors before reaching $\phi=+\infty$, and also the background
charge is $Q=b+\frac{1}{b}$. Note that this exponential potential also breaks
the momentum conservation which its effect during black hole evolution and
island entanglement can be seen. In Liouville thery, the correlation functions
just depend on $b$ and the momenta, therefore the interpolations between the
saddles of entanglement and evolution of island, also just depend on these two
parameters.

For several generalized dilaton theories, in [36], the island formulation for
higher dimensional spherical black holes with asymptotically flat spaces have
been studied. By correcting the black hole solution that is being used, the
authors found that the island formulation can always solve the information
paradox, including for the case of “Liouville black hole solution”, and the
island always appear barely outside of the horizon. There, the island
structures for several charged black holes have also been studied. Specially,
for the case of charged dilaton black hole with the $4d$ metric

$\displaystyle
ds^{2}=-r^{2}\left(1-\frac{2M}{r^{2}}+\frac{Q_{c}^{2}}{4r^{4}}\right)dt^{2}+\left(1-\frac{2M}{r^{2}}+\frac{Q_{c}^{2}}{4r^{4}}\right)^{-1}dr^{2}+r^{2}(dx^{2}+dy^{2}),$
(5.32)

the “effective” $2d$ model would be

$\displaystyle ds^{2}=-H(r)dt^{2}+r^{2}H(r)^{-1}dr^{2},\ \ \ \ X=r^{2},\ \ \ \
\ H(r)=r^{2}\left(1-\frac{2M}{r^{2}}+\frac{Q_{c}^{2}}{4r^{4}}\right),$ (5.33)

and then their results for the position of island match with the result of
[37].

In the late times, the island appear barely outside of the horizon, while in
the early times of radiation the island is “inside” the black hole horizon as
it moves towards the horizon gradually by the Hawking radiation. So the
islands are not stationary but in fact they effectively move by the effect of
the momentum of radiation. The speed of the movement of the island through the
process of radiation and at each step can be studied using the momenta of
Hawking quanta at each step.

One way to understand the movement of island inside the black hole is using
the renormalization flow. So, for the metric of the form

$\displaystyle
ds^{2}=e^{2A(\rho)}[-f(\rho)^{2}dt^{2}+d\vec{x}^{2}]+d\rho^{2},$ (5.34)

the a-function would have the form of

$\displaystyle
a_{T}(\rho)=\frac{\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)\ell_{P}^{d-1}}\left[\frac{f(\rho)}{A^{\prime}(\rho)}\right]^{d-1},$
(5.35)

where the speed of the movement of the island toward the horizon from the
inside, for the general black hole, would be proportional to the derivative of
$a_{T}$ as

$\displaystyle
v_{\text{island}}\propto\frac{da_{T}}{d\rho}=\frac{(d-1)\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)\ell_{P}^{d-1}}\frac{f(\rho)^{d-2}}{A^{\prime}(\rho)^{d}}\times[f^{\prime}(\rho)A^{\prime}(\rho)-f(\rho)A^{\prime\prime}(\rho)].$
(5.36)

Figure 19: Penrose diagram with island configuration for two sided black hole.

Figure 20: The behavior of a-function versus $r$ for Schwarzschild black hole.
Figure 21: The behavior of the a-function versus $q$ and $m$, at $r=2$, for
Schwarzschild black hole.

The behavior of the $a_{T}$ which determines the behavior of the
renormalization flow versus $m$ and $q$ are shown in figure 21, where one
could notice that the mass decreases the flow and the charge would increase
it, which also agrees with the expectation that the mass can slow down the
movement of the island and charge can increase its velocity, as another
evidence that the velocity of the island moving inside of the black hole
toward the horizon is proportional to $a_{T}$.

Note also that there is another velocity dubbed “entanglement velocity”,
$v_{E}$, which has an instantaneous bound $|v_{E}(t)|\leq 1$.

This effect could also be analyzed from the geometric perspective and the $3d$
point of view similar to [1]. The shape of the figure 22 demonstrates that for
Liouville theory the results are compatible as well.

Figure 22: The relations between the entanglement entropy of the island and
the parameters of the Liouville potential, where we fix
$y_{a}^{-}=y_{+}^{a}=2,c=2$.

The Wess-Zumino-Witten (WZW) model and the Liouville theory are also related
to each other by a quantum Drinfeld-Sokolov reduction of the former. Also,
since the correlation functions of the Euclidean $SL_{2}(\mathbb{R})$ WZW
model (dubbed $H_{3}^{+}$ model), the $2d$ black hole $SL_{2}/U_{1}$ and the
theories which continuously interpolate between Liouville and $H_{3}^{+}$
model, could be written in terms of the correlation functions of Liouville
theory, one would expect that the behavior of the island in these theories
would be the same, with no way of distinction. Also, Liouville theory with
$c\geq 25$ can be mapped exactly to some log-correlated random energy models
with a random potential that is logarithmically correlated. The correlation
functions then would get mapped to the correlation functions of the Gibbs
measure of the particle, where in $2d$, it becomes the Gaussian free field
model. Again, we expect that the entanglement entropy cannot distinguish such
theories and island would behave the same way.

For large $c$, i.e, $c=25$, we did not detect a particular specific change,
and only for bigger $c$s, close to zero, a wall is taking shape which is
noticeable after $c=5$. For $c=25$, its plot is shown in figure 23.

Figure 23: The relations between the entanglement entropy of the island and
the parameters of the log-correlated random energy models, for large central
charge and for $y_{a}^{-}=y_{+}^{a}=2,c=25$. The oscillatory wall starts to
appear around $c=5$.

The effects of $b$ on the black hole evolution in Liouville theory, can also
be seen from the disk partition function of Liouville [38] with an FZZT brane
boundary as well, which has the relation

$\displaystyle Z(\mu_{B})^{M}\sim\mu^{\frac{1}{2b^{2}}}\cosh\frac{2\pi s}{b},$
(5.37)

where $\mu_{B}(s)=\kappa\cosh 2\pi bs$ is the FZZT brane parameter, and
$\kappa\equiv\frac{\sqrt{\mu}}{\sqrt{\sin\pi b^{2}}}$. The plot is shown in
figure 24, where again one can see that increasing $b$ would reduce the
partition function.

Figure 24: The relations between the partition function and $b$ in Liouville
theory, where we fix $s=2$.

As shown in [35], the Liouville gravity amplitude and JT gravity partition
functions are also related to each other by a double scaling limit, i.e, $b\to
0$ where the boundary length go to infinity, i.e,
$\ell\sim\frac{\ell_{JT}}{\kappa b}\to+\infty$. Setting $\beta$ to a constant
value such as 2 or $10^{-6}$, one could see the difference between the
entanglement entropy of the island in these two limits in figure 25. Note that
here $C_{0}^{L}$ is the constant coming from the integral which we can vary,
and $\mu$ is the factor in the Liouville potential. The limit of $b\to 0$ is
also related to the genus zero of Weil-Petersson volume.

These results can also be checked from the group interpretations, i.e, BF
theory for the JT gravity, and WZW model for the Liouville theory.

Figure 25: In the left, the entanglement entropy with island in Liouville
gravity, with parameters $c=2$ and $\beta=2$ is shown, and in the right we set
$\beta=10^{-6}$ and $c=2$, where the transformation to a smoother JT theory
can be observed.

In [1], using partial dimension reduction from $3d$ to JT, the authors
concluded that the black hole energy decays exponentially. For the non-
evaporating black holes, they found

$\displaystyle E=-\frac{\Phi_{r}}{8\pi
G}\left\\{\tau,t\right\\}=\frac{2\pi^{2}}{\beta^{2}}\frac{\Phi_{r}^{0}}{8\pi
G}\equiv E_{0},$ (5.38)

while for the evaporating case which the dilaton behaves as
$\tilde{\Phi}_{r}=2\pi\ell\alpha(\tilde{t})\equiv\Phi_{r}^{0}\alpha(\tilde{t})$
and $\alpha(\tilde{t})$ decreases from 1 to 0, the temperature of the $2d$
black hole remains the same, which is directly inherited from the $3d$ BTZ
black hole.

In the evaporating case where the dividing line between the JT gravity and CFT
part of BTZ moves, the equation of motion changes and the dilaton becomes time
dependent as
$\tilde{\Phi}_{r}=\Phi_{r}^{0}\alpha(\tilde{t})=\Phi_{r}^{0}\left(1-\frac{A}{2}\tilde{t}\right)$,
where $\frac{A}{2}=\epsilon\frac{c}{6}\frac{G}{\Phi_{r}^{0}}$ is the
evaporation rate, and the energy decreases as
$\frac{d\tilde{E}}{d\tilde{t}}=-\epsilon\frac{c}{24\pi}\frac{2\pi^{2}}{\beta^{2}}=-E_{0}\frac{A}{2}$.
Using this picture, the partition function can also be time dependent and the
time evolution would induce the interpolation between the JT and Liouville in
the IR and modify the boundary in the UV case.

So considering the UV corrections of [35], and by changing the dimensional
reduction parameter $\alpha$, the evaporation of black hole changes from the
exponential behavior to a power law, specifically this could be observed in
the model of the geometric evaporation.

As for the case of Liouville and JT models, by interpreting $\ell=\beta$ as
inverse temperature, the partition function would be

$\displaystyle Z(\beta)\sim\int_{\kappa}^{\infty}dEe^{-\beta E}\rho_{0}(E),\ \
\ \ \
\rho_{0}(E)=\sinh\left(\frac{1}{b^{2}}\text{arccosh}\frac{E}{\kappa}\right),$
(5.39)

where in the IR limit, i.e, $E=\kappa+\epsilon$, it gives the JT regime, as it
can also be seen from figure 18. In the UV, the Cardy scaling limit of JT
gravity is being modified from $\rho_{0}(E)\sim e^{2\pi\sqrt{E}}$ into power-
law $\rho_{0}(E)\sim E^{1/b^{2}}$. The thermodynamic saddle would also be
$\sqrt{E^{2}-\kappa^{2}}=\frac{1}{b^{2}\beta}$, where in the IR gives the
first law of JT black hole, i.e, $\sqrt{E_{JT}}\sim\beta^{-1}$ and in the UV
shows that the boundary of the bulk geometry is being modified, and it would
not be asymptotically AdS anymore, which has not been considered in [1].
Therefore, for the full solutions, such as the case of fast rotations and a UV
complete theory, to derive the Page curve, this modifications from exponential
to power law should be considered, i.e, additional term should be added to the
Schwarzian action $S=\frac{1}{8\pi G}\int dt\Phi_{r}\\{\tau,t\\}$ which
becomes dominant in the UV limit.

###  5.3 The $3d$ Wess-Zumino-Witten model

Now we would like to find the corresponding parameters for the WZW model in
the $3d$ case, and the corresponding behavior of the islands using this model.

In general, qualitatively one could notice that $\sqrt{h}\phi K$ in $2d$ JT
model corresponds to $\chi A_{0}$ in $2d$ BF model and the term $\sqrt{g}\phi
R$ in JT would correspond to the term $\chi F$ in the BF model. Also, the
Liouville momentum which is a continuous parameter labeled by $k$ is related
to continuous irreducible representation of $SL(2,\mathbb{R})$ labeled by $R$.

Another way to see the connections could be done through the study of [39],
using the deformations of each theory. The defect in the $2d$ JT model (such
as conical defects or wormholes) is related to the deformation of the $1d$
Schwarzian theory where the reparametrization mode is integrated over
different co-adjoint orbits of the Virasoro group. Geometrically, there are
two parameters, $\Phi_{h}$ which is the horizon area operator, and $L(\gamma)$
which is the geodesic length operator, which then as shown in [39] is
structurally related to the deformation of the particle-on-a-group quantum
mechanics where a “chemical potential” or a “magnetic monopole” is being
added. These chemical potentials then would change the partition functions the
way derived in [39].

Note that the effects of these defects in BF theory are similar to the case of
confining backgrounds with the wall at the end of the geometry which can
increase the number of saddles for the mixed system, i.e, for the two
symmetric subsystem, it changes the number of phases from two to four as shown
in [9, 10, 4].

For the partition function of the $3d$ case instead of the $2d$ JT gravity, we
could consider the $SL(2,\mathbb{R})$ WZW model as

$\displaystyle S_{\text{WZW}}[g]=\frac{k_{L}}{8\pi}\int d\tau
d\sigma\sqrt{-h}\text{Tr}(\partial_{a}g^{-1}\partial^{a}g)+k_{L}\Gamma_{\text{WZ}}[g],$
$\displaystyle\Gamma_{\text{WZ}}[g]=-\frac{1}{12\pi}\int\epsilon^{abc}\
\text{Tr}\ (\partial_{a}g\ g^{-1}\partial_{b}g\ g^{-1}\partial_{c}g\ g^{-1}),$
(5.40)

where $k_{L}$ is generally a complex number and is the level on the Riemann
surface $\Sigma$. However, for the compact manifolds we have
$k_{L}\in\mathbb{Z}$, and by comparing with the JT gravity, it could be seen
that $k_{L}$ corresponds to the entropy of black hole at zero temperature.

The fields of this model, $g(z,\bar{z})$, are matrices which are the
(faithful) representation of the Lie-group G, where $g(z,\bar{z})$ is a map
from the manifold to the group, as $g:S^{2}\longrightarrow G$.

The left and right currents of this theory would be defined as

$\displaystyle J_{-}=k_{L}g^{-1}\partial_{-}g,\ \ \ \ \ \ \ \
J_{+}=-k_{L}\partial_{+}gg^{-1},$ (5.41)

and the worldsheet stress tensor is

$\displaystyle
T_{\pm\pm}=\frac{1}{2k_{L}}\text{Tr}(J_{\pm}J_{\pm})\sim-\frac{k_{L}}{4}m^{2},$
(5.42)

where $m$ is the mass of a point-like probe particle. Then, according to the
first law of entanglement entropy, this energy momentum tensor would cause the
shift in the $S_{EE}$.

Note also that in string theory, the $SL(2,\mathbb{R})/U(1)$ gauged WZW model
would be interpreted as the Witten’s $2d$ Euclidean black hole. The spectrum
and partition function of this $3d$ BTZ black hole have been derived in [40,
41]. This partition function for $\text{AdS}_{3}$ has been found as [40]

$\displaystyle
Z=4\sqrt{\tau_{2}}(k_{L}-2)^{\frac{3}{2}}\int^{1}_{0}d^{2}s\int^{1}_{0}d^{2}t\frac{e^{\frac{2\pi}{\tau_{2}}(\text{Im}(s_{1}\tau-
s_{2}))^{2}}}{|\vartheta_{1}(s_{1}\tau-s_{2}|\tau)|^{2}}$
$\displaystyle\times\sum_{m,w,m^{\prime},w^{\prime}\in\mathbb{Z}}\zeta\begin{bmatrix}w+s_{1}-t_{1}\\\
m+s_{2}-t_{2}\end{bmatrix}(k_{L})\ \zeta\begin{bmatrix}w^{\prime}+t_{1}\\\
m^{\prime}+t_{2}\end{bmatrix}(-k_{L}),$ (5.43)

where the free boson conformal blocks that appear in this expression is
defined as

$\displaystyle\zeta\begin{bmatrix}w\\\
m\end{bmatrix}(k_{L})=\sqrt{\frac{k_{L}}{\tau_{2}}}\text{exp}\left(-\frac{\pi
k_{L}}{\tau_{2}}|w\tau-m|^{2}\right),$ (5.44)

and the $\vartheta_{1}$ function is

$\displaystyle\vartheta_{1}(v|\tau)=\sum_{p\in\mathbb{Z}}e^{\pi
i\tau(p+\frac{1}{2})^{2}+2\pi i(v+\frac{1}{2})(p+\frac{1}{2})}.$ (5.45)

By taking $q=e^{2\pi i\tau}$, it can be written as an infinite product as

$\displaystyle\vartheta_{1}(v|\tau)=-2q^{1/8}\sin\pi
v\prod_{p=1}^{\infty}(1-e^{2i\pi v}q^{p})(1-q^{p})(1-e^{-2i\pi v}q^{p}).$
(5.46)

In [41], this partition function has been expanded as

$\displaystyle Z$
$\displaystyle=8i(k_{L}-2)\sum_{w,w^{\prime},n,n^{\prime}\in\mathbb{Z}\
\&N,\bar{N}}\delta_{n,n^{\prime}}\int_{0}^{1}dt_{1}\Bigg{\\{}\int_{0}^{\infty}ds\rho(s)\text{exp}\left[-2\pi\tau_{2}\left(\frac{2s^{2}+1/2}{k_{L}-2}\right)\right]$
$\displaystyle+\sum_{q,\bar{q}}\delta_{n,\bar{q}-q}\text{exp}\left[-2\pi\tau_{2}\left(\frac{-2j(j+1)}{k_{L}-2}\right)\right]_{\frac{1}{2}<-j=\frac{q+\bar{q}}{2}+\frac{k_{L}}{2}(w-t_{1})<\frac{k_{L}-1}{2}}\Bigg{\\}}$
$\displaystyle\times\text{exp}[2\pi
i\tau_{1}(n(w-t_{1})+n^{\prime}(w^{\prime}+t_{1})+N-\bar{N})]$
$\displaystyle\times\text{exp}\left[-2\pi\tau_{2}\left(\frac{n^{2}}{2k_{L}}-\frac{n^{\prime
2}}{2k_{L}}+\frac{k_{L}(w-t_{1})^{2}}{2}-\frac{k_{L}(w^{\prime}+t_{1})^{2}}{2}-\frac{3k_{L}}{12(k_{L}-2)}+N+\bar{N}\right)\right],$

where the density of states is

$\displaystyle\rho(s)=\frac{1}{\pi i}\text{Re}\left[\sum_{q,\bar{q}}^{\ \ \ \
+}\frac{\delta_{n,\bar{q}-q}}{2is+q+\bar{q}+1+k_{L}(w-t_{1})}-\sum_{q,\bar{q}}^{\
\ \ \ \ -}\frac{\delta_{n,\bar{q}-q}}{2is+q+\bar{q}-1+k_{L}(w-t_{1})}\right].$
(5.47)

Similar to [1], we can connect the field $g$ in $3d$ WZW model, (which is a
function of the $2d$ Riemann surface $\Sigma$, onto a Lie group $G$, i.e,
$g:\Sigma\to G$) to the $2d$ JT gravity model as

$\displaystyle\Phi=2\pi\alpha g,$ (5.48)

where $\Phi$ is the dilaton in the JT gravity model, and $\alpha\in(0,1]$ is a
parameter which controls the partial reduction. Note that here $\Phi$ in the
$2d$ case is a scalar while $g(z,\bar{z})$s are matrices.

When the level parameter $k_{L}$ is a positive integer, the affine Lie algebra
is the unitary highest weight representations which would be the dominant
integral. This dominant part then can be connected to the $2d$ JT part easier.
This representation decomposes into finite-dimensional sub-representations
with respect to the subalgebra spanned by each root. If the level is non-
integer, the Lie group $SL(2,\mathbb{R})$ is non-compact and its homotopy
group, $\pi_{3}(SL(2,\mathbb{R}))$, is trivial.

In [1], the time dependence of black hole evolution has been controlled by the
partial reduction of the $3d$ case which divides the geometry into the black
hole part and the bath section. The energy of the black hole decreases
linearly when the diving line moves slowly. From the lower $2d$ dimension
point of view, i.e, JT case, some of the degrees of freedom would be hided in
the one dimension higher, i.e, the bath which could be tracked through the
partition function.

For the case of reduction from $\text{AdS}_{3}$ which is a solution of
“Einstein gravity” to $2d$ JT gravity, in [1], the following relation has been
used

$\displaystyle
K^{(3)}=h^{\mu\nu}K_{\mu\nu}=K^{(2)}+h^{\varphi\varphi}K_{\varphi\varphi},$
(5.49)

which would lead to the relation

$\displaystyle h^{\varphi\varphi}K_{\varphi\varphi}=-\frac{1}{\ell}.$ (5.50)

In this case, for the connection between WZW and JT we can write

$\displaystyle-\frac{k_{L}}{8\pi}K^{(3)}(\gamma^{-1}\partial^{\mu}\gamma)-\frac{k_{L}}{24\pi}\epsilon^{ijk}K^{(3)}(\gamma^{-1}\partial_{i}\gamma)=-\frac{S_{0}}{2\pi}\left(K^{(2)}+h^{\varphi\varphi}K_{\varphi\varphi}\right).$
(5.51)

For the BTZ case, similarly one would have
$h^{\varphi\varphi}K_{\varphi\varphi}=-\frac{1}{\ell}$.

So in the WZW model, instead of $\log k$ which specifies the number of
orthonormal states, we should take $\log k_{L}$, which is related to $S_{0}$
of JT gravity.

Then, the question is that whether in this case, contributions from other
modes than the s-waves, or the effects such as greybody factor or Schwinger
effects, should be considered or not. If one assumes that the distance between
points of one’s interest is much larger than the length scale of their size,
then one can only consider the s-wave contribution and use the $2d$ CFT
results.

In [42], the off-shell partition function for the chiral-gravity which is
related to PSL(2, R) Chern-Simons theory has been calculated, and they found
that there is a scaling limit where their partition function reduces to JT
gravity using the equivariant localization. The level $k$ in chiral gravity is
proportional to the dimensionless ratio $\ell_{\text{AdS}}/G_{N}$ and the dual
CFT has a left-moving central charge $c=24k$. The higher genus partition
functions become complicated due to the oscillatory relation to the Newton’s
constant, which lead to the result that gravity can indeed gives non-smooth
contribution to the partition function.

The higher genus corrections can actually construct the structure of $1/N$
corrections in the dual symmetric orbifold CFT. These higher genus and multi-
boundary amplitudes are actually very important in understanding very late-
time correlators behavior and the Page curve.

In [42], the large $k$ limit has also been implemented, where the
approximation using the Weil-Petersson volumes of moduli space were used.
There, they also found that higher genus partition functions get “oscillatory
behavior” in $k$, which its signature in island formalism and Hawking
radiation could be detected. This behavior for instance could make the islands
move from inside of the black hole toward slightly outside of it in the late
times. The gravity in this regimes would not give the smooth contributions to
the CFT partition function or the spectral form factor, where its effects can
be injected in the lower dimensional JT gravity. The speed of the islands
movement toward the horizon and the outward from it would then be directly
related to this oscillation in $k$. In models of [43], it has also been found
that since CGHS/RST model are defined on fixed topology where there is no
replica wormholes, but by summing over “topologies”, as the bottom-up
modification, one can get the island rule formulation for the entropy of
radiation, demonstrating again the effects of these higher genus topologies on
the appearance and dynamics of the island. This is also related to the result
of [43] in the form of the relation
$S_{\text{grav}}=\frac{c}{6}\rho=\Omega+\text{cosntant}$, where $\rho$ is the
conformal factor, and $\Omega$ is the field in the solution of CGHS/RST, and
the piece $\frac{c}{6}\rho$ comes from the Weyl transformation and from the
$2d$ bath CFT, showing the theory knows the “generalized entropy” gives the
correct result.

The effects of these higher genus and various topologies can also be
investigated by quantum error correction methods such as Petz map and modular
flow, since in the JLMS and entanglement wedge reconstruction, the equality of
relative entropy with and without the island as
$S_{\text{rel}}(\rho_{R}|\sigma_{R})=S_{\text{rel}}(\rho_{R\cup
I}^{\text{semi-cl}}|\sigma_{R\cup I}^{\text{semi-cl}})$ is a main basis as the
states in the island $I$ are encoded in the radiation part $R$. The dynamics
of the island would not change this structure and can be investigated using
this conserved quantity.

Another piece of evidence comes from the monotonicity of mutual information
between the island and the region $R$ as [43]

$\displaystyle\frac{d}{d\lambda_{+}}I_{\text{mat}}(I,R)\geq 0,$ (5.52)

meaning the correlations between the island region and radiation increases,
which is compatible with the picture that the island move from inside of the
horizon toward outside increasing this mixed correlations. The above relation
is part of a more general relation
$\pm\frac{d}{d\lambda_{\pm}}S_{\text{gen}}(I)\geq 0$

To get a better handle on this effect, we plot the behavior of $Z_{0}(\beta)$
and $Z_{1}(\beta)$ as shown in figures 26 and 27.

Figure 26: First order of the partition function, $Z_{0}(\beta)$, versus
$\beta$, for k=5, with the even or odd number of terms, $m$ [42]. Figure 27:
The next order of the partition function, $Z_{1}(\beta)$, versus $\beta$, for
k=5, [42].

In the case of [42], the limits where they could reach to JT gravity is
$k\to\infty$ (related to small $\hbar$ limit) and $\beta\to 0$ (small thermal
radius), while $\beta_{JT}=\frac{1}{k\beta}$ is being fixed. The reason then
that the oscillatory behavior in $k$ which depends on the whole properties of
the topology and the background geometry, and the local dynamical behavior of
the island depends on such a non-local parameter is due to the Haar-invariance
and the delocalized quantum information. So, similar to the Gaussian Unitary
Ensemble (GUE) model for Hamiltonian used in [44], the island dynamics which
also depends on the long-time physics of a local system will similarly have
become delocalized like the GUE models. So the onset of the approximate Haar-
invariance which can be characterized using the “$k$-invariance” [44] can
determine if the correlation is local or completely non-local at each step,
and when/if sharp phase transitions occurs.

Then, in [31], in the coset manifold and in the $(\varphi,v,\bar{v})$
coordinates, the action of the strings propagation has been written as

$\displaystyle S=\frac{k_{b}}{\pi}\int
d^{2}z(\partial\varphi\bar{\partial}\varphi+(\partial\bar{v}+\bar{v}\partial\varphi)(\bar{\partial}v+v\bar{\partial}\varphi)).$
(5.53)

The one loop string amplitude in the background of BTZ black hole has been
derived in [31] as

$\displaystyle
Z=\int_{\mathcal{F}_{0}}\frac{d^{2}\tau}{4\tau_{2}^{2}}Z_{\text{BTZ}}(r_{\pm})Z_{gh}Z_{int},$
(5.54)

where the partition functions of the ghost and internal sectors are

$\displaystyle Z_{gh}=\tau_{2}|\eta(\tau)|^{4},$ $\displaystyle
Z_{int}=(q\bar{q})^{-\frac{c_{int}}{24}}\sum_{h}d(h,\bar{h})q^{h}\bar{q}^{\bar{h}},$
$\displaystyle
Z_{BTZ}=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\sum_{w,m}\frac{e^{-\pi\frac{k_{b}}{\tau_{2}}r_{+}^{2}|m-w\tau|^{2}+\frac{2\pi}{\tau_{2}}\text{Im}(\bar{U}_{m,w})^{2}}}{|\theta_{1}(\bar{U}_{m,w},\tau)|^{2}},$
(5.55)

where $d(h,\bar{h})$ denotes the degeneracy of the states of the internal CFT
and $q=e^{2\pi i\tau}$ is the elliptic norm. Also, here the holonomy $\bar{U}$
is

$\displaystyle\bar{U}_{m,w}=(r_{-}-ir_{+})(m-w\tau).$ (5.56)

The $Z_{BTZ}$ part could be expanded as [31]

$\displaystyle
Z_{BTZ}=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\frac{1}{|\eta|^{6}}\sum_{w,mr,\bar{r}}S_{r}S_{\bar{r}}e^{2\pi
imr-(r-\bar{r})}e^{2\pi mr_{+}(1+r+\bar{r})}$ $\displaystyle
q^{-(r+\frac{1}{2})w(r_{-}-ir_{+})}\bar{q}^{-(\bar{r}+\frac{1}{2})w(r_{-}+ir_{+})}e^{-\pi\frac{k_{b}}{\tau_{2}}r_{+}^{2}|m-w\tau|^{2}+\frac{2\pi}{\tau_{2}}\text{Im}(\bar{U}_{m,w})^{2}}.$
(5.57)

Then, this “twisted partition function” of BTZ case can also be compared with
the $2d$ JT case and then the Page curve can be derived, as the parameters
$r_{+}$, $w$ and $\eta$ become time dependent. We leave the full calculations
to the future works.

It worths to mention here that, for studying the movement of the island, the
conformal welding problem [45] could also be considered, but instead one may
use the average BCFT model of [46], to study this problem. The more complete
picture in fact can come from considering the time-dependence of bulk
primaries $\phi_{i}$ and their two-point function, as they have “finite”
distance from each other. For doing that the higher orders of the following
relations need to be considered.

$\displaystyle\phi_{i}(z)\sim\sum_{I}(2\mathfrak{T}_{z})^{h_{I}-h_{i}-\bar{h}_{i}}C_{iI}^{a}\phi_{I}(\mathfrak{R}_{z})+....,$
(5.58)

$\displaystyle\langle\phi_{i}\phi_{i}\rangle_{\text{disk}}\langle\phi_{i}\phi_{i}\rangle_{\text{disk}}=\sum_{p}(C_{iip})^{2}(C_{p\mathbb{I}}^{a})^{2}\mathcal{F}_{ii}^{ii}(h_{p}|z)\mathcal{F}_{ii}^{ii}(h_{p}|z^{\prime}).$
(5.59)

##  6 Mixed correlations and Islands: the view from $1d$ higher

In this section, we examine how to find the effects of island in various mixed
correlation measures, models and examples from $1d$ higher point of view in
the setup of VV [1].

###  6.1 The view of negativity and island from $1d$ higher

It has been proposed in [47], that the holographic entanglement negativity is
half of the Rényi reflected entropy of order $1/2$. In [48, 49, 50], the
entanglement negativity has been calculated for evaporating black holes in the
island setups and in holographic models with a defect brane, demonstrating the
island/$\text{BCFT}_{2}$ or island/defect models. The island formulation for
entanglement negativity has also recently been proposed in [51]. Two proposals
for the island contribution to negativity have been proposed there. The first
proposal involves extremizing the algebraic sum of the generalized Rényi
entropies of order half, and the second one involves extremizing the sum of
the area of a back-reacted brane on the EWCS.

The first proposal could be written as

$\displaystyle\mathcal{E}^{\text{gen}}(A:B)=\frac{1}{2}\Bigg{[}S_{\text{gen}}^{(1/2)}(A\cup
C)+S_{\text{gen}}^{(1/2)}(B\cup C)-S_{\text{gen}}^{(1/2)}(A\cup B\cup
C)-S_{\text{gen}}^{(1/2)}(C)\Bigg{]}$
$\displaystyle\mathcal{E}(A:B)=\text{min}(\text{ext}_{Q^{\prime\prime}}\\{\mathcal{E}^{\text{gen}}(A:B)\\}),$
(6.1)

where $C$ is the system which is between $A$ and $B$ and
$Q^{\prime\prime}=\partial I_{s_{\mathcal{E}}}(A)\cap\partial
I_{s_{\mathcal{E}}}(B)$, and $S_{\text{gen}}^{(1/2)}$ which is the generalized
Rényi entropy of order half has the relation

$\displaystyle S^{(1/2)}_{\text{gen}}(A)=\frac{\mathcal{A}^{(1/2)}[\partial
I_{s}(A)]}{4G_{N}}+S_{\text{eff}}^{(1/2)}(A\cup I_{s}(A)),$ (6.2)

and corresponds to the effective Rényi entropy of order half of the quantum
matter fields coupled to semiclassical gravity.

The second proposal could be written in the most general form as

$\displaystyle\mathcal{E}^{\text{gen}}(A:B)=\frac{\mathcal{A}^{(1/2)}(Q^{\prime\prime}=\partial
I_{s_{\mathcal{E}}}(A)\cap\partial
I_{s_{\mathcal{E}}}(B))}{4G_{N}}+\mathcal{E}^{\text{eff}}(A\cup
I_{s_{\mathcal{E}}}(A):B\cup I_{s_{\mathcal{E}}}(B))$
$\displaystyle\mathcal{E}(A:B)=\text{min}(\text{ext}_{Q^{\prime\prime}}\\{\mathcal{E}^{\text{gen}}(A:B)\\}).$
(6.3)

Then, using these two proposals, the contribution of island to entanglement of
negativity in the setup of VV [1] can be studied. This can easily be done by
replacing the each term of entropy in 6.1 with the one from $1d$ higher point
of view of VV and then derive the Page curve in that setup.

###  6.2 Islands in Kaluza-Klein black holes

In [52], the behaviors of the islands in spherically symmetric Kaluza-Klein
(KK) black holes have been studied. The KK charge denoted by $Q$ would deform
the black holes from the Schwarzschild form. This charge slightly extends the
boundary of the island in late times, increases the Page time by a factor of
$(1+Q/r_{h})$ and the scrambling time by a factor of $(1+Q/r_{h})^{1/2}$. The
charge $Q$ also reduces the surface gravity and the Hawking temperature. This
charge also depends on the $\omega$ in the higher genus partition functions,
found for the off-shell action in [42].

With the action of Kaluza-Klein theory with the Lagrangian

$\displaystyle\mathcal{L}=\sqrt{-g}\left(R-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{4}e^{\sqrt{2(D-1)/(D-2)}}F^{2}\right),$
(6.4)

after compactifying one of the spatial coordinates on a circle
$\mathbb{S}^{1}$, a scalar and a vector field would be emerged.

The metric anasatz would also be

$\displaystyle
d\hat{s}^{2}_{D+1}=e^{2\alpha\phi}ds_{D}^{2}+e^{2\beta\phi}(dz+A_{\mu}dx^{\mu})^{2},$
(6.5)

where

$\displaystyle\alpha^{2}=\frac{1}{2(D-1)(D-1)},\ \ \ \ \ \ \
\beta=-(D-2)\alpha,$ (6.6)

which unlike the non-extremal Reissner–Nordström black hole has only one
horizon.

The $4d$ Kaluza-Klein black hole which is a solution of the Lagrangian would
be

$\displaystyle
ds^{2}=-W(r)dt^{2}+\frac{dr^{2}}{W(r)}+H^{1/2}r^{2}d\Omega^{2},$ (6.7)

where

$\displaystyle W(r)=f(r)/\sqrt{H(r)},\ \ \ \ f(r)=1-\frac{r_{h}}{r},\ \ \ \ \
H(r)=1+\frac{Q}{r}.$ (6.8)

For the case without the island, the entanglement entropy of one interval
$[b_{-},b_{+}]$ is

$\displaystyle S_{\text{mat}}([b_{-},b_{+}])=\frac{c}{3}\log d(b_{-},b_{+}),$
(6.9)

where $d(x,y)$ can be found from

$\displaystyle d^{2}(x,y)=|\Phi(x)\Phi(y)(U(x)-U(y))(\bar{U}(x)-\bar{U}(y))|,$
(6.10)

and $\Phi$ are the elements of the Euclidean metric as
$ds^{2}=\Phi^{2}dUd\bar{U}$.

For the case with the island, the entanglement entropy for two intervals would
be

$\displaystyle S_{\text{mat}}(\mathcal{R}\cup
I)=\frac{c}{3}\log\left[\frac{d(a_{+},a_{-})d(b_{+},b_{-})d(a_{+},b_{+})d(a_{-},b_{-})}{d(a_{+},b_{-})d(a_{-},b_{+})}\right].$
(6.11)

Then, using the KK metric above, the entropy $S$ for each case can be found
and the Page curve be derived. Note that another source of the movement of the
island inside of the black hole toward the horizon is the tachyonic Kaluza
Klein modes discussed in [53] which makes the lower-dimensional supergravity
masses to break the Breitenlohner-Freedman (BF) bound which then in the full
11 or 10 dimensions makes the vacuum perturbatively unstable. Specifically,
the “brane-jet instability” of $SO(3)\times SO(3)$ vacuum [54] would thrust
the islands. This thrust is due to the net repulsive force of probe branes
arising in certain areas of compactification manifold, which is the
consequence of varying warp factors of the $11d$ solution, and this
instability would be triggered by higher KK modes. These modes then change the
structure of the islands.

From the picture of AdS/BCFT and the island creation on the end of the world
brane (EOW), [33], as shown in figure 28, it could understood better how the
the tensions between the branes and these brane-jet instabilities can push the
islands from inside of the black hole in the early stages to outside of the
horizon in the later times. By looking from $1d$ higher, the various
discontinuities in the entanglement contours observed in the presence of the
island and the boundary, which can be explained by the localization-
delocalization interpolations of entanglement, can be better observed.

Figure 28: Island position on the end of the world brane relative to the bulk
and boundary. The tension between the branes, as explained in the text, would
push the island from inside of the black hole toward the outside.

From the behavior of the island and unitarity and resolving the information
paradox it might be possible to explain why the tachyons can only appear in
the symmetric representations $(\mathbf{k},\mathbf{k})$ of $SO(3)\times
SO(3)$, or other general theories.

Another evidence for the existence of an island inside of the black hole can
come from the second solution for the set of differential equations, found in
[55] using the micro-canonical path integral, which could be written as

$\displaystyle V_{K}^{(2)}\approx V_{K}^{B}+\frac{(L^{2}+2\mu
U_{K}^{B}V_{K}^{B})}{3\mu U_{K}^{B}},\ \ \ \
U_{K}^{(2)}\approx\frac{5U_{K}^{B}}{3}+\frac{1}{3\mu V_{K}^{(B)}},$ (6.12)

where $(U_{K},V_{K})$ are elements of the Kruskal coordinate

$\displaystyle ds^{2}=-e^{2\rho(V_{K},U_{K})}dU_{K}dV_{K},\ \ \ \
e^{2\rho}=\frac{4\mu}{(1+\frac{\mu}{L^{2}}U_{K}V_{K})^{2}}.$ (6.13)

Figure 29: The structure of the second (“un-physical” from the point of view
of an outer observer) solution of [55] which can lie inside of the eternal
black hole horizon.

This solution from the point of view of an observer outside of the black hole
horizon is unphysical. But from the point of view of an observer inside the
horizon, it can be a physical solution of the set of equation of motion of
quantum corrected version of JT gravity and it can be new saddle. This saddle,
in specific cases, such as charged extremal case, can move toward the outside
of the black hole horizon, leading again to the result that during the
evaporation, island transfer from inside toward slightly outside of the
horizon.

The radial position of the quantum extremal surface in the Schwarzchild
coordinates is

$\displaystyle
r_{\text{QES}}=L\sqrt{\mu}\left(\frac{1-\frac{\mu}{L^{2}}U_{K}V_{K}}{1+\frac{\mu}{L^{2}}U_{K}V_{K}}\right)\approx
r_{H}\left(-1+\frac{18\mu L^{2}U_{K}^{B}V_{K}^{B}}{L^{4}+19\mu
L^{2}U_{K}^{B}V_{K}^{B}+25\mu^{2}(U_{K}^{B}V_{K}^{B})^{2}}\right).$ (6.14)

The structure of this solution is shown in figure 29. Note that the
thermodynamic for the static and nested wedge observer versus accelerating
observers, would also be different, leading to the result that for another
coordinate system and sets of observers, the above solution can indeed become
physical.

###  6.3 Islands for charged dilaton black holes and its view from higher $d$

In this subsection, the effects of dilaton on island from higher $d$ point of
view is examined. In [56], the island prescription for $4d$ linear dilaton
black holes with planar horizon has been studied. These black holes asymptotes
to the linear dilaton background. In principle, these black holes are actually
a two-dimensional Witten black hole with two additional free bosons. This
picture in the string frame would help us more to understand how information
from higher dimensions will be encoded in the lower dimensions. The picture is
also that the entanglement entropy would have a “running” behavior, as it
changes along the RG flow with respect to the two-dimensional worldsheet
length scale. In fact, it has been shown that as the islands force the entropy
to decrease, the dilaton runs toward the IR point.

In [56], also, it has been found that without an island, the entropy behaves
asymptotically as

$\displaystyle S=S_{\text{matter}}\sim\frac{c}{3}\frac{t_{b}}{r_{h}},$ (6.15)

which grows linearly with time, while when an island is included in the
computation one would get

$\displaystyle
S\approx\frac{k^{-2}e^{-\sigma}}{2\alpha^{\prime}}\Big{|}_{\text{horizon}}\approx\frac{r_{h}^{2}}{2G_{N}},$
(6.16)

which is independent of time and is proportional to twice of the Bekenstein-
Hawking entropy. Here $k$ is a constant with mass dimension $1$, and in each
coordinate of the $x$ and $y$, the scale is $k^{-1}$.

There, it has been shown that without the island the entropy behaves as
$S\approx\frac{c}{3}\frac{t_{b}}{r_{0}}e^{-a\lambda}$, which blows up in the
IR. However, when an island is included, the entropy scales as
$S\approx\frac{r_{h}^{2}}{2G_{N}}=\frac{r_{0}^{2}}{2G_{N}}e^{2a\lambda}$,
which has a decreasing behavior toward IR.

In [37], the island prescription for charged linear dilaton black holes has
also been studied, both for the non-extremal and extremal cases. It has been
found that the Page time is universal for all of the different models that
they have been studied and it would be $t_{\text{Page}}=\frac{3}{\pi
c}\frac{S_{\text{BH}}}{T_{H}}$. Now the question would be if this quantity is
still universal looking from higher dimensions or it would change.

The action considered in [37] is four-dimensional dilaton action with a $U(1)$
gauge field in the Einstein frame as

$\displaystyle I=\frac{1}{16\pi G_{N}}\int
d^{4}x\sqrt{g}\left(R-\frac{1}{2}(\partial\sigma)^{2}+4k^{2}e^{\sigma}-\frac{1}{4}e^{\gamma\sigma}F_{\mu\nu}F^{\mu\nu}\right),$
(6.17)

where $k$, $\gamma$ and $\sigma$ are constant and $\sigma$ is a scalar field.

For the case without an island, and by calculating the length of the geodesic,
for the solution of the charged dilaton black hole, the entropy of the matter
fields would be

$\displaystyle
S_{\text{matter}}=\frac{c}{3}\log\left[2f(b)e^{\kappa_{+}r^{*}(b)}\cosh
k_{+}t\right],$ (6.18)

where for $r_{+}\ll b$ would be

$\displaystyle S_{\text{matter}}=\frac{c}{3}\log(2\cosh
k_{+}t)\simeq\frac{c}{3}\kappa_{+}t,$ (6.19)

which increases linearly with time. From the results of [37], the diagrams
shown in figure 30 can be found.

Figure 30: Diagrams coming from the results of relations found in [37]. The
left figure is $S_{\text{matter}}$ versus $t_{a}$, or $t_{b}$, for the non-
extremal charged dilaton black hole, with the presence of an island, the
middle is the plot of the generalized entropy, $S_{gen}$, versus $t_{a}$ or
$t_{b}$, while one of them is being fixed for the same case, and finally the
right one comes from generalized entropy $S_{\text{matter}}$, versus $r_{h}$
for the “extremal charged dilaton” black hole case.

An interesting issue here is the claim in [37], which argued that the
description for the extremal case cannot be obtained from the continuous limit
of the non-extremal case, and without the island no correct result for the
extremal case can be obtained. This is because the geodesics distances that
need to be calculated without the island, i.e, eq. 5.1 of [37] is ill-defined
due to the divergences at $r=0$, or one could say since the origin is not part
of the manifold, or the periodicity of the Euclidean time is not fixed. In
[57], the Page curve for Garfinkle-Horowitz-Strominger dilaton black holes has
also been discussed. They also found that the charge $Q$ would have a
significant impact on Page time. In addition, they found that there is some
peculiarity for the extremal case and there the Page time either diverges or
would vanish, as the extremal black hole don’t behave quite physically.

Note that as explained in [58], the entropy of an extremal black hole is zero
from semi-classical method, and $S=\pi\rho^{2}/G$ from the string theory
microstate counting, and the lack of the entropy from in the first approach
could be explained by the creation of a separate dimension within the black
hole. The island can be considered inside of this separated region going to
other dimension or the whole region can be considered as the island of this
case. So this way correct form of the Page curve for the extremal case would
be related to the question of how information from higher $d$, for instance
$4d$ Schwarzschild or Reissner–Nordstrom black holes would be encoded in the
lower $2$ near horizon of black holes, leading to the pattern of entanglement
entropy and island prescription for the extremal case.

###  6.4 Extremality from higher $d$

In [59] the motivation also was to address the issue of finite entropy of
extremal charged black hole (or black brane) at zero temperature, which can be
addressed by the perspective we got here from looking from $1d$ higher to BH
observables. For studying the problem of entropy of 3-charged extremal black
holes, the authors studied the response to the probe Fermionic operators in
different limits. Then, the fluctuations around the Fermi-surfaces in these
different limits of charges and frequencies have been examined.

One interesting outcome of that work was that the bottom-up approach to
holography could give wrong or un-physical results and it would be necessary
to address this problem in the top-down full string theory or supergravity
backgrounds. For the case of $N=4$ SYM which has three independent chemical
potentials (charges), when all the three charges are on, the horizon is non-
singular and we have the issue of finite entropy in zero temperature. However,
there are limits of charges (or approaching the extremality) in which this
zero-temperature entropy is absent which gives insight into the nature of this
entropy.

For $N=4$, when one charge is absent, (and for simplicity taking the other two
charges equal) the entropy would vanish as the horizon have zero-area and is
singular. Then, the Fermion responses were studied in this limit. The
interesting physical fact here is that this 2-charged case is the solution of
$5D$ gauged supergravity background which comes from the reduction of type IIB
supergravity background on $\text{AdS}_{5}\times S^{5}$. This corresponds to a
specific state in $N=4$ SYM where in this full theory, it was found that the
masses depend on the running scalar field (dilaton) which diverges at the
horizon/singularity. So for understanding the Fermionic responses in this
limit of extremal black brane, one should consider these couplings and running
dilatons from higher $d$ to lower $d$.

The other interesting finding of that work is that for the extremal two-charge
black hole an energy scale “$\lambda$” exists that within this energy from the
Fermi surface, fluctuations would not decay. In fact, there is a special
sector with large density of states near the horizon that for generic three
charge case is not gapped and extends down to Fermi surface leading to the
non-zero entropy and unstable ground state but in the two-charge case, this
sector is gapped which cannot get excited by energies less that “$\lambda$”
and so is stable. This gap removes the large entropy in zero temperature and
changes the mechanism of Fermionic fluctuations and also the behavior of all
the zero modes. As it has been studied further in [1], this idea is also
related to the correct calculation of Euclidean saddles in gauge/gravity and
the island prescriptions [60], with correct boundary conditions needed for the
in-falling particles.

In [59], they also noted that this stability might indicate that the fermionic
fluctuations have no self-interaction which is intrinsic and that could be a
large-N effect. Turning even a small charge $q$ other than the equal two
charges would suddenly remove the gap. So in the regular three-charge case,
the near-horizon and small-$\omega$ limits don’t commute which indicates there
are two regions, the inner and outer regions which are patched together. The
inner region has the geometry of $\text{AdS}_{2}\times R^{3}$, where the outer
region could help us solve our issue. It can be shown that in a two-charge
case, this geometry can be lifted to a smooth $\text{AdS}_{3}\times R^{3}$ in
six dimensions, leading to a well-behaved geometry, and the final stability.
The Kaluza-Klein charge of the reduction is actually that $q_{1}$ that is
being turned off in the two-charged case. The gap also can be understood in
this picture as the minimum energy needed to turn a momentum vector which has
a fixed amount of compact momentum time-like, which is also related to the
amount of energy needed to push the island toward out of horizon. The angular
component of the metric has the role of the dilaton which is the Kaluza-Klein
charge of the reduction. The authors also suggest that the field theory
explanations of this gap could also be constructed using the emergent non-
chiral Virasoro algebra in the infrared which again points out to a universal
feature.

So this behavior is very similar to the Page phase transition of the black
hole, where here the charge $q_{1}$ controls the phase transition. This gap is
indeed like the exchanges between the saddles of quantum extremal surfaces
(QES) that jumps from the empty surface to a surface inside of the black hole
horizon. Hence, this gap is like the island that appears after the Page time
while here, when the charge $q_{1}$ is being turned on, the phase transition
can be initiated. This issue can also be solved using a doubly holographic
model for the gap inside the near horizon of the black brane.

Thus, this two inner and outer regions that Gubser discussed is like the black
hole and the bath in JT gravity where the $\text{AdS}_{2}$ part is like the
black hole and the bath is in the higher dimensional uplifted case that is
being controlled by the charge $q_{1}$. The entropy of $\text{AdS}_{2}$ would
transfer to the higher dimensional part, making the whole geometry stable
then. By turning on and off the $q_{1}$ charge like the case for black hole
which connects the black hole interior to radiation, here too, the higher
dimension geometry can connect the $\text{AdS}_{2}$ with whatever needed to
elevate it to the $\text{AdS}_{3}$. After turning on the charge $q_{1}$, the
black hole interior becomes part of the entanglement wedge of (bath or
radiation of BH for the case here) the geometry which is needed to be added to
the $\text{AdS}_{2}$ part to elevate it to an $\text{AdS}_{3}$, so these
islands decrease the large entropy seen for the $\text{AdS}_{2}$ case. Note
also that these islands are inside of the black hole horizon and so here the
second geometry would be inside of the black hole, while $\text{AdS}_{2}$ is
the outer region.

In conclusion, the $\text{AdS}_{2}$ is like the black hole case and the bath
is like the remainder of the geometry which we call here
$G_{in}=\text{AdS}_{3}-\text{AdS}_{2}$. This is the remaining internal
geometry which by turning on the charge $q_{1}$ would separate $G_{in}$ from
$\text{AdS}_{2}$, making the near horizon unstable. When the charge $q_{1}$
becomes bigger, more of the entropy goes into the $\text{AdS}_{2}$, when
$q_{1}=0$, the $G_{in}$ or the bath, appears inside of the black hole, and
when $q_{1}$ becomes more negative, the size of the bath increases further.

##  7 Measurable observables from $1d$ higher

In this section, various real and physical observables and the mechanisms that
through dimensional reduction can be analyzed in the setup island formulation
will be discussed.

###  7.1 Quantum Fluctuations

So some of the results from the dimensional reductions can in fact be tested
using measurable quantities.

For instance, recently, in [61], using the dimensional reduction from $4d$
Einstein-Hilbert action to $2d$ JT gravity, the quantum fluctuations and the
uncertainty in the time of a photon traveling from tip-to-tip of a causal
diamond, in the $4d$ flat Minkowski case has been calculated. This is related
to our study here connecting Hawking radiation in the higher $d$ case to the
one dimensional lower.

Specifically, near the horizon, the causal development of a region in flat
Minkowski spacetime can be described by the $2d$ dilaton theory which can be
described by the JT action, and there the quantum fluctuations and uncertainty
would lead to the metric fluctuations, where this uncertainty in the higher
$4d$ case would lead to the uncertainty of the photon’s travel time from tip-
to-tip of a causal diamond.

In principle, the dialton field in the lower dimensional case is related to
the area of the transverse directions. The modular fluctuations discussed in
[61] which has been proposed to be experimentally observable could actually be
the reason that the island region moves inside the black hole toward the
horizon. These fluctuations could be discussed from the higher dimensions
point of view as in [61]. The main point is to figure out how the island
fluctuations would be Weyl rescaled between the dimensions. As explained in
[61], the dynamics of the dilaton would control the size of $S^{2}$ part and
also the relative position of the horizon (and therefore the island) with
respect to the boundary. Since this dilaton has its own effective hydrodynamic
description, it could effectively explain the dynamics of the island inside
the horizon.

In [61], the near-horizon quantum dynamics of $4d$ Einstein gravity has been
studied from $2d$ JT gravity model. The metric fluctuations in JT action,
leads to the quantum uncertainty of the position of the horizon. There, an
important relation for our story had also been found, i,e,