we are considering with vanishing Hawking flux might not be the classical
saddle that we assumed to have a horizon. The saddle point is modified by the
quantum effects since the back-reaction makes a significant difference in the
geometry of the saddle. We should move to the correct saddle by including the
quantum corrections.

To begin with, let us assume the following ansatz for the back-reacted
geometry

$\begin{split}ds^{2}&=-F(r)e^{2\epsilon\varphi(r)}dt^{2}+\frac{dr^{2}}{F(r)}\\\
&=F(r)e^{2\epsilon\varphi(r)}\bigg{(}-dt^{2}+\frac{dr^{2}}{F(r)e^{2\epsilon\varphi(r)}}\bigg{)}\\\
&=F(r)e^{2\epsilon\varphi(r)}(-dt^{2}+dr^{\ast 2}).\end{split}$ (201)

where

$F(r)=1-\frac{r_{0}}{r}+\frac{\epsilon m(r)}{r}\equiv F_{0}(r)+\frac{\epsilon
m(r)}{r},$ (202)

with $\epsilon=\frac{G_{N}\hbar}{24\pi}$. We have also defined the tortoise
coordinate $r^{\ast}$

$r^{\ast}\equiv\int\frac{1}{F(r)e^{\epsilon\varphi}}dr.$ (203)

Now we introduce the Eddington-Finkelstein coordinates

$u=t-r^{\ast},\quad v=t+r^{\ast}.$ (204)

We have

$ds^{2}=-F(r)e^{2\epsilon\varphi(r)}dudv.$ (205)

While the ansatz for the metric already implies the effect of back-reaction
sourced by $\epsilon$ is small, we shall see concretely in the following how
contradiction from the back-reaction equations arises and the assumption about
the existence of a horizon fails when we go nearer to the would-be horizon.

With the metric ansatz, the semi-classical Einstein equations sourced by the
quantum stress tensor read

$-\epsilon F_{0}(r)m^{\prime}(r)=2G_{N}\langle T_{tt}\rangle,$ (206)
$2\epsilon rF_{0}(r)\varphi^{\prime}(r)=2G_{N}\bigg{(}F_{0}(r)\langle
T_{rr}\rangle+\frac{\langle T_{tt}\rangle}{F_{0}(r)}\bigg{)}.$ (207)

Starting from the stress tensors we derived in (68) and (69), we work out

$\displaystyle\langle T_{rr}\rangle=$
$\displaystyle\frac{\hbar}{12\pi}\left(1-\frac{r_{0}}{r}\right)^{-2}\left[\frac{3r^{2}_{0}}{8r^{4}}-\frac{r_{0}}{2r^{3}}\right]+\frac{\hbar}{8\pi
r^{2}}\ln\left(1-\frac{r_{0}}{r}\right)$ (208) $\displaystyle-\frac{\hbar
r_{0}}{12\pi r^{3}}\left(1-\frac{r_{0}}{r}\right)^{-1},$

$\displaystyle\langle T_{tt}\rangle=$
$\displaystyle\frac{\hbar}{12\pi}\left[\frac{3r^{2}_{0}}{8r^{4}}-\frac{r_{0}}{2r^{3}}\right]+\frac{\hbar}{8\pi}\left(1-\frac{r_{0}}{r}\right)^{2}\frac{1}{r^{2}}\ln\left(1-\frac{r_{0}}{r}\right)$
(209) $\displaystyle+\frac{\hbar r_{0}}{12\pi
r^{3}}\left(1-\frac{r_{0}}{r}\right).$

We then find that there is already an inconsistency in the equation of motion:
when $r=r_{0}$, the LHS of the first equation in (206) vanishes, while
$\langle T_{tt}\rangle$ does not, this means $m^{\prime}\left(r\right)$ has to
be divergent in $r=r_{0}$. Indeed, by solving the solutions for the functions
$m(r)$ and $\varphi(r)$, one can explicitly check the two functions blow up at
the horizon, and we expect such behavior to persist beyond one-loop order.
This is due to the fact that the back-reaction is strong and the ansatz (201)
that is perturbative in $\epsilon$ is no longer applicable in the near-horizon
regime.

In other words, (201) assumes that we get a smooth geometry with a horizon
when we turn off the back-reaction in the $\epsilon\to 0$ limit. However,
since the stress tensor becomes divergent in the near-horizon regime, where
the quantum fluctuation can no longer be viewed as a small perturbation, we
expect a dramatic change in the near-horizon structure. Therefore, we need to
analyze the near-horizon geometry starting from a generic ansatz that works
for arbitrary two-dimensional geometry.

In the following, we shall apply the ansatz

$\begin{split}ds^{2}&=-C(r)dt^{2}+\frac{C(r)}{H^{2}(r)}dr^{2}\\\
&=-e^{2\rho\left(r\right)}{\rm d}u{\rm d}v,\end{split}$ (210)

where we will express everything in terms of a generic conformal factor
$\rho(r)$. In the following, we discuss the conditions for the existence of a
horizon when back-reaction is included. We have
$C\left(r\right)=e^{2\rho\left(r\right)}$, and suppose the horizon exists and
is specified by $r=r_{H}$, which leads to the fact that

$C\left(r_{H}\right)=0,\quad\rho\left(r_{H}\right)=-\infty,\quad\rho^{\prime}\left(r_{H}\right)=\infty.$
(211)

This is essentially the reason why the perturbative analysis in $\epsilon$
breaks down: when we work in the regime where $\left(\epsilon\rho^{\prime
2}\right)\simeq O\left(1\right)$, it can no longer be viewed as perturbation
around classical geometry, instead, it changes the “classical background”
significantly. Therefore, we solve the semi-classical Einstein equation
sourced by $\langle T_{ab}\rangle$ in the following

$e^{-2\phi}\\{2\nabla_{a}\nabla_{b}\phi-2\nabla_{a}\phi\nabla_{b}\phi+g_{ab}[3(\nabla\phi)^{2}-2\Box\phi]\\}-g_{ab}=2G_{N}\left\langle
T_{ab}\right\rangle.$ (212)

The $tt$-component gives

$\displaystyle
e^{2\rho}-2rH(r)H^{\prime}(r)+H^{2}(r)\left(2r\rho^{\prime}(r)-1\right)=$
$\displaystyle
2\epsilon\left[\frac{H(r)}{r}H^{\prime}(r)\left(r\rho^{\prime}(r)+6\right)\right.$
(213)
$\displaystyle+\frac{H^{2}(r)}{r^{2}}\left(r^{2}\rho^{\prime\prime}(r)-r^{2}\rho^{2}(r)\right.$
$\displaystyle\left.\left.-6r\rho^{\prime}(r)+6\rho(r)-6\right)\right],$

while the $rr$-component gives

$1-\frac{e^{2\rho(r)}}{H^{2}(r)}+2r\rho^{\prime}(r)=2\epsilon\left[\frac{6\rho(r)}{r^{2}}-\frac{6\rho^{\prime}(r)}{r}-\rho^{\prime
2}(r)\right].$ (214)

From (214) we can solve $H\left(r\right)$ in terms of $\rho\left(r\right)$ as

$H(r)=\pm\frac{e^{\rho(r)}r}{\sqrt{D(r)}},$ (215)

where we have introduced

$D(r)=r^{2}+2r^{3}\rho^{\prime}(r)+2\epsilon\left(r^{2}\rho^{\prime
2}(r)+6r\rho^{\prime}(r)-6\rho(r)\right).$ (216)

Substituting either the positive or negative root of $H\left(r\right)$ back
into (213), and eliminating some overall factors, we find the equation of
motion in terms of $\rho(r)$

$\displaystyle 0=$ $\displaystyle
144\epsilon^{2}\rho^{2}(r)-12r\epsilon\rho(r)\left[\rho^{\prime}(r)\left(3r^{2}+17\epsilon+3r\epsilon\rho^{\prime}(r)\right)-r\epsilon\rho^{\prime\prime}(r)\right]$
(217)
$\displaystyle+r^{2}\left[6\epsilon+2\rho^{\prime}(r)\left(r^{2}+6\epsilon+r\epsilon\rho^{\prime}(r)\right)\left(r+\rho^{\prime}(r)\left(r^{2}+6\epsilon+r\epsilon\rho^{\prime}(r)\right)\right)\right.$
$\displaystyle\left.+\left(r^{4}+11r^{2}\epsilon+36\epsilon^{2}+r\epsilon\left(r^{2}+6\epsilon\right)\rho^{\prime}(r)\right)\rho^{\prime\prime}(r)\right].$

This differential equation is extremely complicated and cannot be solved
exactly. However, the method of dominant balance suffices for us to analyze
the dominant solutions in different regimes.

As a consistency check, let us first assume the existence of a horizon at
$r=r_{H}$, and work in the regime where $\epsilon^{-1/2}\simeq\rho^{\prime
2}\simeq\rho^{\prime\prime}$. We are still in the perturbative framework in
terms of $\epsilon$ and we can omit terms that are higher order in $\epsilon$,
that is

$2r^{5}\rho^{\prime}+2r^{6}\rho^{\prime
2}+r^{6}\rho^{\prime\prime}+O(\epsilon)=0.$ (218)

The solution can therefore be chosen as

$2\rho(r)=\ln{\bigg{(}1-\frac{r_{H}}{r}\bigg{)}}+{\rm const},$ (219)

which reproduces the Schwarzschild form. The scale $r_{H}$ that appears in the
above solution can be defined in the asymptotic boundary as the ADM mass of
the whole spacetime. However, the naïve solution above breaks down as we get
nearer to $r_{H}$, such that $\rho^{\prime\prime},\rho^{\prime
2}\gg\epsilon^{-1}$. In this case, the following two terms dominate

$2r^{4}\epsilon^{2}\rho^{\prime
4}+r^{5}\epsilon\rho^{\prime}\rho^{\prime\prime}+\dots=0,$ (220)

where terms in the dots are with fewer derivatives in $\rho$. We find then in
this case $\rho$ caps off in a very sharp region, with

$\rho\left(r\right)=\frac{r_{H}}{2}\sqrt{\frac{\pi}{\epsilon}}{\rm
Erfi}\left(\sqrt{\ln\left(r/r_{H}\right)}\right),$ (221)

where $r_{H}$ is an integration constant where we have defined it to be the
position where $\rho\left(r_{H}\right)=0$. Namely, at $r=r_{H}$ the geometry
actually caps off and forms a smooth cone. There is no horizon at all, and the
physical portion of spacetime only contains the part $r\geq r_{H}$.

We further comment that for the aforementioned analysis to work, we should
keep $r$ very close to $r_{H}$ such that

$\frac{r-r_{H}}{r_{H}}\ll\frac{\epsilon}{r_{H}^{2}}.$ (222)

Within this small region, the back-reaction of the Boulware modes is so strong
that the geometry deviates significantly from the one with a horizon. Instead,
it forms a static and spherically symmetric quantum star and ends at a
definite value $r=r_{H}$ of the radius. We interpret the point with $r=r_{H}$
as the center of the star, which is similar to the origin of the polar
coordinates. In this way, we do not need to impose extra boundary conditions
there. We present a numerical verification of the claim in Figure 3 and Figure
4 below.131313Earlier works such as Ho:2017joh  described the resulting
geometry as a wormhole-like structure with $r_{H}$ being the effective radius
of the throat. This scenario is related to our discussion by identifying the
two sides of a wormhole together with the asymptotic boundary.

Figure 3: A plot of the numerical solution of (217) (blue), Schwarzschild
black hole solution (219) (green) and near $r_{H}$ solution (221) (red). We
have set $\epsilon=0.01,r_{H}=0.1$ and the constant in (219) to be 4.95. It is
clear that the numerical solution matches (221) in a narrow near $r_{H}$
regime very well, which justifies our approximation. It asymptotes to a
Schwarzschild black hole when the scale $r$ is significantly larger than
$\epsilon^{(1/2)}$ .

Figure 4: The left graph zooms into the regime where the numerical solution
deviates from the two approximate solutions. It is a regime deep inside the
bulk but still away from $r_{H}$. The geometry suffers from the back-reaction
sourced by the stress tensor of quantum matter and fails to form a horizon.
The right graph zooms into the regime near $r_{H}$, where the stress tensor
component is significant compared with the curvature. We see in this case the
numerical solution matches the near $r_{H}$ solution (221) well.

Now let us imagine the journey of an infalling observer starting at spatial
infinity. When $r\gg r_{H}$, she feels to be in the vacuum state exterior to a
black hole. Then as she moves deeper and closer to $r_{H}$, but with
$r-r_{H}\gg\epsilon^{1/2}$, the quantum fluctuation becomes significant, which
locates at what she thinks to be the putative horizon. She would not encounter
anything unusual and will reach the other side of the star if she manages to
survive and pass through $r=r_{H}$.

We end this section by commenting on the fact that the Einstein tensor for the
solution discussed here satisfies $G^{t}_{t}=G^{r}_{r}=O(1)+O(\sqrt{r-r_{H}})$
near $r\simeq r_{H}$. This means the prescribed solution and the definition of
the Boulware state do not require Planck scale physics and is consistent with
the low-energy effective dynamics of gravity specified by the semi-classical
Einstein equations.

##  Appendix C Island Computation in the Hartle-Hawking State

In this appendix, we detail the computation of the Page curve following the
island prescription for the fine-grained entropy, namely

$S_{\text{gen}}(R)=\text{min}_{I}\bigg{\\{}\text{ext}_{I}\bigg{[}\frac{\text{Area}(\partial
I)}{4G_{N}}+S_{\text{matter}}(I\cup R)\bigg{]}\bigg{\\}}.$ (223)

We will first consider the no-island case where we shall reproduce Hawking’s
prediction on a monotonically increasing entropy. Note that in this case, the
area term is absent, therefore we identify the matter entropy part as our
fine-grained entropy, which is given by

$S_{{\rm
matter}}=\frac{1}{12}\ln\frac{\left(V_{R}-V_{L}\right)^{2}\left(U_{R}-U_{L}\right)^{2}}{\delta^{4}e^{-2\rho_{R}}e^{-2\rho_{L}}}.$
(224)

Here we denote $L/R$ as the left/right asymptotically flat region of
spacetime. We define the coordinates on the cut-off surface to be
$\left(t,b^{\ast}\right)$. $\delta$ is the UV cut-off. Plugging the definition
of the coordinates

$\begin{gathered}V_{R}=\frac{1}{\kappa}e^{\kappa\left(t+b^{\ast}\right)},\quad
U_{R}=-\frac{1}{\kappa}e^{-\kappa\left(t-b^{\ast}\right)},\\\
V_{L}=-\frac{1}{\kappa}e^{\kappa\left(-t+b^{\ast}\right)},\quad
U_{L}=\frac{1}{\kappa}e^{-\kappa\left(-t-b^{\ast}\right)}.\end{gathered}$
(225)

where the conformal factor is the same in both regions as $b$ is the same,
hence we know $\rho_{L}=\rho_{R}$. We find

$S_{\text{matter }}(t)=\frac{1}{6}\ln\frac{4F(b)\cosh^{2}(\kappa
t)}{(\kappa\delta)^{2}e^{-2\epsilon\varphi(b)}}.$ (226)

We regularize the UV divergence by demanding that initially the entropy is
zero, namely $S_{{\rm mattter}}\left(0\right)=0$, and this fixes

$\delta^{2}=\frac{4F(b)e^{2\epsilon\varphi(b)}}{\kappa^{2}}.$ (227)

Putting the definition of $\delta$ back into the formula, we get

$S_{{\rm matter}}=\frac{1}{3}\ln\left(\cosh\left(\kappa t\right)\right).$
(228)

At early times of the evaporation, namely $\kappa t\ll 1$, the entropy behaves
as

$S_{{\rm matter}}\simeq\frac{1}{6}\left(\kappa t\right)^{2},$ (229)

while the more interesting case is at late times when $\kappa t\gg 1$, we find

$S_{{\rm matter}}\simeq\frac{1}{3}\kappa t+{\rm const}.$ (230)

The fine-grained entropy of the radiation increases monotonically in time,
which is in agreement with Hawking’s original calculation corresponding to the
no-island case. The above result is general for the two-dimensional dilaton
gravity model, with the only difference being that now we have a quantum
corrected $\kappa$. We shall see in the following how the island prescription
restores unitarity.

Figure 5: The Penrose diagrams of the two-sided black hole. The left and right
asymptotically flat regions are denoted as L and R. We take the states
associated with the Hawking radiation to be represented by $R$ where the cut-
off surfaces are written simply as $b$. Similarly, for the island region $I$
and the associated quantum extremal surfaces $\partial I=a$. By
complementarity, the calculation of the matter entropy involves two disjoint
intervals $B$ on the two sides.

With island, we need to consider the entropy formula for two disjoint
intervals with (See Figure 5)

$S_{\text{matter}}=\frac{1}{6}\ln{\frac{d^{2}_{12}d^{2}_{23}d^{2}_{14}d^{2}_{34}}{\delta^{4}d^{2}_{24}d^{2}_{13}e^{-\rho_{1}}e^{-\rho_{2}}e^{-\rho_{3}}e^{-\rho_{4}}}},$
(231)

where $d^{2}_{ij}=(V_{i}-V_{j})(U_{i}-U_{j})$. Suppose the position of the
island is given by $(t^{\prime},a^{\ast})$ on the right patch (and similarly
$(-t^{\prime},a^{\ast})$ on the left patch), we will be able to find the exact
position by varying with respect to $a^{\ast}$. Now we have the following
relations

$V_{Rb}=\frac{1}{\kappa}e^{\kappa(t+b^{\ast})},\quad
U_{Rb}=-\frac{1}{\kappa}e^{-\kappa(t-b^{\ast})},$ (232)
$V_{Lb}=-\frac{1}{\kappa}e^{\kappa(-t+b^{\ast})},\quad
U_{Lb}=\frac{1}{\kappa}e^{-\kappa(-t-b^{\ast})},$ (233)
$V_{Ra}=\frac{1}{\kappa}e^{\kappa(t+a^{\ast})},\quad
U_{Ra}=-\frac{1}{\kappa}e^{-\kappa(t-a^{\ast})},$ (234)
$V_{La}=-\frac{1}{\kappa}e^{\kappa(-t^{\prime}+a^{\ast})},\quad
U_{La}=\frac{1}{\kappa}e^{-\kappa(-t^{\prime}-a^{\ast})},$ (235)

By using the fact that at late times $t\simeq t^{\prime}$

$\frac{d_{23}^{2}d_{14}^{2}}{d_{24}^{2}d_{13}^{2}}\rightarrow 1,\quad
d_{12}=d_{34},\quad\rho_{1}=\rho_{4},\quad\rho_{2}=\rho_{3},$ (236)

the matter entropy term becomes

$S_{\text{matter
}}=\frac{1}{3}\left(\rho_{a}+\rho_{b}\right)+\frac{2}{3}\ln\left(e^{\kappa
b^{\ast}}-e^{\kappa a^{\ast}}\right)-\frac{2}{3}\ln\kappa\delta.$ (237)

Adding to the area term (with a factor of $2$ because we have two
asymptotically flat regions), we write the fine-grained entropy as

$S_{\text{gen }}=\frac{2\pi
a^{2}}{G_{N}\hbar}+\frac{1}{3}\left(\rho_{a}+\rho_{b}\right)+\frac{2}{3}\ln\left(e^{\kappa
b^{\ast}}-e^{\kappa a^{\ast}}\right)-\frac{2}{3}\ln\kappa\delta.$ (238)

We extremize the entropy with respect to $a^{\ast}$

$\partial_{a^{\ast}}S_{\text{gen }}=\frac{4\pi
a}{G_{N}\hbar}\frac{da}{da^{\ast}}+\frac{1}{3}\frac{d\rho(a)}{da}\frac{da}{da^{\ast}}-\frac{2}{3}\frac{\kappa}{e^{\kappa\left(b^{\ast}-a^{\ast}\right)}-1}=0.$
(239)

By definition, we know ${\rm d}a/{\rm
d}a^{\ast}=F\left(a\right)e^{\epsilon\varphi\left(a\right)}$, and by noting
that $\epsilon=\frac{G_{N}\hbar}{24\pi}$, we get

$\left[a+2\epsilon\rho^{\prime}\left(a\right)\right]F(a)e^{\epsilon\varphi(a)}=4\epsilon\frac{\kappa}{e^{\kappa\left(b^{\ast}-a^{\ast}\right)}-1}.$
(240)

In the case of an eternal black hole, we expect to find the quantum extremal
surface to be near but outside the horizon Almheiri:2019yqk . Without loss of
generality, we consider the near-horizon expansion where we take the island
position to be

$a=r_{H}+x,\quad x\ll r_{H}.$ (241)

The expansion on the LHS of (240) becomes

$\text{LHS}\simeq\bigg{(}r_{H}+2\epsilon\frac{d\rho}{dr}\bigg{|}_{H}\bigg{)}\bigg{(}\frac{dF}{dr}e^{\epsilon\varphi(r_{H})}\bigg{)}\bigg{|}_{H}x=2\kappa
x(r_{H}+2\epsilon\rho^{\prime}|_{H}),$ (242)

by dropping terms with $F(r_{H})=0$. We have also used the definition of the
surface gravity where $(\frac{dh}{dr}e^{\epsilon\varphi(r)})|_{H}=2\kappa$.
For the RHS of (240), we consider the cut-off surface to be far away from the
horizon, hence only $a^{\ast}=r^{\ast}(a)$ is relevant in the expansion. Note
that $r^{\ast}(r_{H})\to-\infty$, we have the following two choices for
$e^{\kappa a^{\ast}}$

  * •

The leading order piece of $e^{\kappa a^{\ast}}$ is an $O(1)$ constant, which
means that the island position is at a small fixed location away from the
horizon. In this case, we do not need to expand on the RHS of (240) and we
expect the correction coming from $x$ is $O(\epsilon)$. This can be verified
by substituting (242) into (240)

$x=\frac{2\epsilon}{(r_{H}+2\epsilon\rho^{\prime}|_{H})[e^{\kappa(b^{\ast}-a^{\ast})}-1]}\approx\frac{2\epsilon}{r_{0}[e^{\kappa(b^{\ast}-a^{\ast})}-1]}+O(\epsilon^{2}).$
(243)

  * •

The leading order piece of $e^{\kappa a^{\ast}}$ is $O(x)$, which means the
island is extremely close to the horizon and they are nearly identical. In
this case, the correction coming from $x$ will be of $O(\epsilon^{2})$, in
agreement with Hashimoto:2020cas ; Matsuo:2020ypv ; Djordjevic:2022qdk . Let
us consider the expansion

$\displaystyle e^{2\kappa a^{\ast}}$ $\displaystyle\approx$ $\displaystyle
e^{2\kappa r^{\ast}(r_{H})}+2\kappa\bigg{(}e^{2\kappa
r^{\ast}}\frac{dr^{\ast}}{dr}\bigg{)}\bigg{|}_{H}x$ (244) $\displaystyle=$
$\displaystyle 2\kappa
xe^{1+\epsilon\alpha(r_{H})}\frac{e^{-\epsilon\varphi(r_{H})}}{r_{H}h^{\prime}(r_{H})}=\frac{x}{r_{H}}e^{1+\epsilon\alpha(r_{H})}.$
(245)

Note that we have used the following near-horizon expansion for $r^{\ast}(r)$
Djordjevic:2022qdk

$r^{\ast}(r)\approx\frac{1}{2\kappa}\bigg{[}\frac{r}{r_{H}}+\ln{\bigg{(}\frac{r}{r_{H}}-1\bigg{)}}+\epsilon\alpha(r)\bigg{]},$
(246)

where $\alpha(r)$ denotes the terms that do not diverge at the horizon. We can
work out this relation easily by considering the near-horizon expansion of the
tortoise coordinate

$\displaystyle r^{\ast}$ $\displaystyle\equiv$
$\displaystyle\int\frac{e^{-\epsilon\varphi(r)}}{h(r)}dr$ (247)
$\displaystyle\approx$
$\displaystyle\int\frac{e^{-\epsilon\varphi(r_{H})}-e^{-\epsilon\varphi(r_{H})}(r-r_{H})\epsilon\varphi^{\prime}(r_{H})+\cdots}{h(r_{H})+h^{\prime}(r_{H})(r-r_{H})+\frac{1}{2}h^{\prime\prime}(r_{H})(r-r_{H})^{2}+\cdots}dr$
$\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int\frac{1}{r-r_{H}}\bigg{[}\frac{1-(r-r_{H})\epsilon\varphi^{\prime}(r_{H})+\cdots}{1+\frac{1}{2}\frac{h^{\prime\prime}(r_{H})}{h^{\prime}(r_{H})}(r-r_{H})+\cdots}\bigg{]}dr$
$\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int\bigg{[}\frac{1}{r-r_{H}}-\bigg{(}\frac{1}{2}\frac{h^{\prime\prime}}{h^{\prime}}+\epsilon\varphi^{\prime}\bigg{)}\bigg{|}_{H}+\frac{\epsilon}{2}\bigg{(}\frac{h^{\prime\prime}}{h^{\prime}}\varphi^{\prime}\bigg{)}\bigg{|}_{H}(r-r_{H})\bigg{]}dr$
$\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\bigg{[}\ln{\bigg{(}\frac{r}{r_{H}}-1\bigg{)}}-\frac{1}{2}\frac{h^{\prime\prime}(r_{H})}{h^{\prime}(r_{H})}r-\epsilon\varphi^{\prime}(r_{H})r$
$\displaystyle+\frac{\epsilon}{2}\frac{h^{\prime\prime}(r_{H})}{h^{\prime}(r_{H})}\varphi^{\prime}(r_{H})\bigg{(}\frac{r^{2}}{2}-rr_{H}\bigg{)}+C\bigg{]},$

where we have used $h(r_{H})=0$ and
$2\kappa=h^{\prime}(r_{H})e^{\epsilon\varphi(r_{H})}$. With the following
relation

$-\frac{1}{2}\frac{h^{\prime\prime}(r_{H})}{h^{\prime}(r_{H})}r\approx\frac{r}{r_{H}}-\frac{\epsilon
rr_{H}}{2r_{0}}m^{\prime\prime}(r_{H}),$ (248)

we confirm (246). Hence the RHS of (240) can be expanded as

RHS $\displaystyle\approx$ $\displaystyle 4\epsilon\kappa e^{-\kappa
b^{\ast}}e^{\kappa a^{\ast}}(1+e^{-\kappa b^{\ast}}e^{\kappa a^{\ast}})$ (249)
$\displaystyle=$ $\displaystyle
4\epsilon\kappa\bigg{[}\sqrt{\frac{x}{r_{H}}}e^{\frac{1}{2}(1+\epsilon\alpha(r_{H}))-\kappa
b^{\ast}}+\frac{x}{r_{H}}e^{1+\epsilon\alpha(r_{H})-2\kappa
b^{\ast}}\bigg{]}.$ (250)

We solve for $x$ as

$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{1}{r_{H}}\frac{(\frac{2\epsilon}{r_{H}})^{2}e^{1-2\kappa
b^{\ast}+\epsilon\alpha(r_{H})}}{[1+\frac{2\epsilon}{r_{H}}(\rho^{\prime}|_{H}-\frac{1}{r_{H}}e^{1-2\kappa
b^{\ast}+\epsilon\alpha(r_{H})})]^{2}}$ (251) $\displaystyle\approx$
$\displaystyle\frac{4\epsilon^{2}}{r^{3}_{h}}e^{1-2\kappa
b^{\ast}}+O(\epsilon^{3}).$

Note that $x$ is positive in both cases, which means the island is indeed
outside the horizon. This confirms our initial assumption. We then conclude
this section by comparing the entropy in the no-island and island phases. We
have in the island case the $S_{\text{gen}}$ being

$\displaystyle S_{\text{gen}}(a)$ $\displaystyle=$ $\displaystyle
S_{\text{gen}}(r_{H})+S_{\text{gen}}^{\prime}(r_{H})x+O(x^{2})$ (252)
$\displaystyle\approx$ $\displaystyle S_{\text{gen}}(r_{H})+\frac{4\pi
r_{H}}{G_{N}\hbar}x,$

where

$\displaystyle S_{\text{gen}}(r_{H})$ $\displaystyle=$
$\displaystyle\frac{2\pi
r^{2}_{H}}{G_{N}\hbar}+\frac{1}{3}(\rho_{H}+\rho_{b})+\frac{2}{3}\ln{\frac{e^{\kappa
b^{\ast}}}{\kappa\delta}}$ (253) $\displaystyle=$ $\displaystyle
2\bigg{[}\frac{\pi r^{2}_{H}}{G_{H}\hbar}+\frac{1}{12}\ln{\frac{e^{4\kappa
b^{\ast}}}{(\kappa\delta)^{4}e^{-2\rho_{H}}e^{-2\rho_{b}}}\bigg{]}}.$ (254)

We can see from (252) that if $x\sim O(\epsilon)$, the correction can be
$O(1)$ in $\epsilon=\frac{G_{N}\hbar}{24\pi}$. If $x\sim O(\epsilon^{2})$, the
correction is essentially negligible. Therefore, if we keep only up to the
$O\left(1\right)$ terms of the entropy, we can approximately think of the
island located at the position of the back-reacted horizon. In either case,
the fine-grained entropy at late times is given by

$S_{\text{FG}}=\text{min}\bigg{\\{}\frac{1}{3}\kappa
t,S_{\text{gen}}(a)\bigg{\\}}.$ (255)

The Page time is then the transition time where

$\frac{1}{3}\kappa t_{P}\approx S_{\text{gen}}(a)\implies t_{P}=3\kappa
S_{\text{gen}}(a).$ (256)

##  Appendix D Island Computation in the Unruh State

Similarly, we compute the island position where the generalized entropy is
given by

$S_{{\rm gen}}=\frac{\pi r_{a}^{2}}{G}+S_{{\rm matter}},$ (257)

and $S_{{\rm matter}}$ can be written in $\left(U,v\right)$ coordinate as

$S_{\text{matter
}}=\frac{c}{6}\log\left(U_{a}-U_{b}\right)\left(v_{a}-v_{b}\right)+\frac{c}{12}\left(\kappa_{a}v_{a}+\epsilon\varphi_{a}+\kappa_{b}v_{b}+\epsilon\varphi_{b}\right)+\text{const},$
(258)

where in the following, we will denote
$r_{a}=r(v_{a}),r_{Ha}=r_{H}\left(v_{a}\right),\kappa_{a}=\kappa\left(v_{a},r_{Ha}\right)$,
and similarly for $b$. Here $a$ corresponds to $\partial I$, and $b$ is the
cut-off surface. It is necessary to specify the concrete surface gravity at
$a$ and $b$ because now we are not in a stationary case, and it is likely that
for general extremal island configuration that $v_{a}\not=v_{b}$.

We are mainly interested in the late-time configuration of the island, and the
endpoint of it is expected to be in the near-horizon regime Penington:2019npb
, where the following approximation is applicable

$r\left(v\right)=r_{H}\left(v\right)-Ue^{\kappa
v}+\frac{1}{\kappa}r_{H}^{\prime}+O\left(\epsilon^{2}\right).$ (259)

The physical meaning of $U$ coordinate is the relative deviation for a radial
in-falling null geodesic from the horizon, and $\kappa$ is the surface gravity
at $r_{H}$. The above equation (259) should be understood to work
perturbatively in $\epsilon$, where the horizon position $r_{H}$ and surface
gravity $\kappa$ should all be viewed as functions of $\epsilon$. Therefore we
expect that the time derivatives of $r_{H}$ and $\kappa$ should belong to
$O(\epsilon)$, and (259) is exact only up to $O(\epsilon)$.

Now we start by extremizing the generalized entropy

$\begin{split}\frac{\partial S_{{\rm gen}}}{\partial
v_{a}}&=\frac{r_{a}}{12\epsilon}\left(r_{Ha}^{\prime}-\kappa_{a}U_{a}e^{\kappa_{a}v_{a}}\right)+\frac{c}{6}\frac{1}{v_{a}-v_{b}}+\frac{c}{12}\kappa_{a}+O\left(\epsilon\right)=0,\\\
\frac{\partial S_{{\rm gen}}}{\partial
U_{a}}&=\frac{r_{a}}{12\epsilon}\left(-e^{\kappa_{a}v_{a}}\right)+\frac{c}{6}\frac{1}{U_{a}-U_{b}}+O\left(\epsilon\right)=0,\end{split}$
(260)

where we only keep terms up to $O\left(1\right)$ in $\epsilon$, and used the
fact that $r_{Ha}^{\prime},\kappa_{a}^{\prime}\simeq O\left(\epsilon\right)$,
where the prime denotes the derivative with respect to $v$. Combining the two
equations we find a useful relation between $U_{a}$ and $U_{b}$

$U_{a}=U_{b}\frac{r_{a}r_{Ha}^{\prime}\left(v_{a}-v_{b}\right)+c\epsilon\left(2+\kappa_{a}\left(v_{a}-v_{b}\right)\right)}{r_{a}r_{Ha}^{\prime}\left(v_{a}-v_{b}\right)+c\epsilon\left(2-\kappa_{a}\left(v_{a}-v_{b}\right)\right)}.$
(261)

Since the leading order of the fraction above is $O(1)$ in $\epsilon$, the
equation indicates that $U_{a}$ and $U_{b}$ are at the same order in
$\epsilon$. Plugging (261) back into the extremal equation (260) of
$S_{\rm{gen}}$, we find the solution of $U_{a}$ reads

$U_{a}=\frac{\epsilon
ce^{-\kappa_{a}v_{a}}\left(2+\kappa_{a}\left(v_{a}-v_{b}\right)\right)}{\kappa_{a}r_{Ha}\left(v_{a}-v_{b}\right)}+\frac{r_{Ha}^{\prime}}{\kappa_{a}}e^{-\kappa_{a}v_{a}}+O\left(\epsilon^{2}\right),$
(262)

and the result is manifestly at $O\left(\epsilon\right)$, which is consistent
with the near-horizon approximation (259).

Notice that we have assumed that the endpoint of the cut-off surface
$b=(U_{b},v_{b})$ locates outside the horizon. Let us verify this by plugging
$U_{a}$ in (262) back into (261), which gives

$\begin{split}U_{b}&=\frac{\epsilon
ce^{-\kappa_{a}v_{a}}\left(2-\kappa_{a}\left(v_{a}-v_{b}\right)\right)}{\kappa_{a}r_{Ha}\left(v_{a}-v_{b}\right)}+\frac{r_{Ha}^{\prime}}{\kappa_{a}}e^{-\kappa_{a}v_{a}}+O\left(\epsilon^{2}\right),\\\
r_{b}&=r_{Hb}-\frac{\epsilon
c\left(2-\kappa_{a}\left(v_{a}-v_{b}\right)\right)}{\kappa_{a}r_{Ha}\left(v_{a}-v_{b}\right)}e^{\kappa_{b}v_{b}-\kappa_{a}v_{a}}\\\
&+\left(\frac{r_{Hb}^{\prime}}{\kappa_{b}}-\frac{r_{Ha}^{\prime}}{\kappa_{a}}e^{\kappa_{b}v_{b}-\kappa_{a}v_{a}}\right)+O\left(\epsilon^{2}\right).\end{split}$
(263)

where we have used the definition (259) in deriving the radial position of
$b$. For an evaporating black hole with a shrinking horizon, one always has
$r_{H}^{\prime}<0$ and $\kappa^{\prime}>0$. Then it is clear to deduce from
(263) that the endpoint of the cut-off surface is outside the horizon as long
as $v_{a}-v_{b}<0$. The condition $v_{a}-v_{b}<0$ is imposed because a
timelike surface is not included in the extremization procedure. The fact that
$b$ is always outside the horizon is consistent with our setup.

Finally, let us discuss the location of the island by considering the
difference $r_{a}-r_{Ha}$ with $U_{a}$ given by (262), The result is

$r_{a}=r_{Ha}-\frac{\epsilon
c\left(2+\kappa_{a}\left(v_{a}-v_{b}\right)\right)}{\kappa_{a}r_{Ha}\left(v_{a}-v_{b}\right)}+O\left(\epsilon^{2}\right),$
(264)

An interesting point of the result is that there is no explicit dependence on
the kinematics of the horizon. That is, terms with explicit dependence on time
derivative of $r_{H}$ all cancel among themselves. The contribution from back-
reaction can all be absorbed into quantum corrections of the surface gravity
$\kappa_{a}$. This implies that we do not need to specify a concrete form of
evolution of the horizon when analyzing the position of the island. One can
easily deduce from (264) that for $|v_{a}-v_{b}|>2/\kappa_{a}$, then
$r_{a}<r_{Ha}$ where the island sits inside the horizon, and for
$|v_{a}-v_{b}|<2/\kappa_{a}$, it extends outside the horizon.

## References

  * (1) S. Hawking, _Breakdown of Predictability in Gravitational Collapse_ , _Phys. Rev. D._ 14 (1976) 2460.
  * (2) G. Penington, _Entanglement Wedge Reconstruction and the Information Paradox_ , _JHEP_ 09 (2020) 002 [1905.08255].
  * (3) A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, _The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole_ , _JHEP_ 12 (2019) 063 [1905.08762].
  * (4) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, _The Page curve of Hawking radiation from semiclassical geometry_ , _JHEP_ 03 (2020) 149 [1908.10996].
  * (5) G. Penington, S.H. Shenker, D. Stanford and Z. Yang, _Replica wormholes and the black hole interior_ , 1911.11977.
  * (6) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _Replica Wormholes and the Entropy of Hawking Radiation_ , _JHEP_ 05 (2020) 013 [1911.12333].
  * (7) D.N. Page, _Information in black hole radiation_ , _Phys. Rev. Lett._ 71 (1993) 3743 [hep-th/9306083].
  * (8) D.N. Page, _Time Dependence of Hawking Radiation Entropy_ , _JCAP_ 1309 (2013) 028 [1301.4995].
  * (9) A. Almheiri, R. Mahajan and J.E. Santos, _Entanglement islands in higher dimensions_ , _SciPost Phys._ 9 (2020) 001 [1911.09666].
  * (10) F.F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, _Page Curve for an Evaporating Black Hole_ , _JHEP_ 05 (2020) 091 [2004.00598].
  * (11) T. Anegawa and N. Iizuka, _Notes on islands in asymptotically flat 2d dilaton black holes_ , _JHEP_ 07 (2020) 036 [2004.01601].
  * (12) T. Hartman, E. Shaghoulian and A. Strominger, _Islands in Asymptotically Flat 2D Gravity_ , _JHEP_ 07 (2020) 022 [2004.13857].
  * (13) K. Hashimoto, N. Iizuka and Y. Matsuo, _Islands in Schwarzschild black holes_ , _JHEP_ 06 (2020) 085 [2004.05863].
  * (14) Y. Matsuo, _Islands and stretched horizon_ , _JHEP_ 07 (2021) 051 [2011.08814].
  * (15) X. Wang, R. Li and J. Wang, _Page curves for a family of exactly solvable evaporating black holes_ , _Phys. Rev. D_ 103 (2021) 126026 [2104.00224].
  * (16) J. Tian, _Islands in Generalized Dilaton Theories_ , 2204.08751.
  * (17) W.-C. Gan, D.-H. Du and F.-W. Shu, _Island and Page curve for one-sided asymptotically flat black hole_ , _JHEP_ 07 (2022) 020 [2203.06310].
  * (18) S. Djordjević, A. Gočanin, D. Gočanin and V. Radovanović, _Page Curve for Eternal Schwarzschild Black Hole in Dimensionally-Reduced Model of Dilaton Gravity_ , 2207.07409.
  * (19) M.-H. Yu and X.-H. Ge, _Entanglement Islands in Generalized Two-dimensional Dilaton Black Holes_ , 2208.01943.
  * (20) C.-Z. Guo, W.-C. Gan and F.-W. Shu, _Page curves and Entanglement Islands for the Step-Function Vaidya Model of Evaporating Black Holes_ , 2302.02379.
  * (21) C. Krishnan, _Critical Islands_ , _JHEP_ 01 (2021) 179 [2007.06551].
  * (22) T. Hartman, Y. Jiang and E. Shaghoulian, _Islands in cosmology_ , _JHEP_ 11 (2020) 111 [2008.01022].
  * (23) S.E. Aguilar-Gutierrez, A. Chatwin-Davies, T. Hertog, N. Pinzani-Fokeeva and B. Robinson, _Islands in Multiverse Models_ , _JHEP_ 11 (2021) 212 [2108.01278].
  * (24) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Int. J. Theor. Phys._ 38 (1999) 1113 [hep-th/9711200].
  * (25) S.S. Gubser, I.R. Klebanov and A.M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett._ B428 (1998) 105 [hep-th/9802109].
  * (26) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (27) K. Goto, T. Hartman and A. Tajdini, _Replica wormholes for an evaporating 2D black hole_ , 2011.09043.
  * (28) R. Jackiw, _Lower Dimensional Gravity_ , _Nucl. Phys. B_ 252 (1985) 343.
  * (29) C. Teitelboim, _Gravitation and Hamiltonian Structure in Two Space-Time Dimensions_ , _Phys. Lett. B_ 126 (1983) 41.
  * (30) C.G. Callan, Jr., S.B. Giddings, J.A. Harvey and A. Strominger, _Evanescent black holes_ , _Phys. Rev. D_ 45 (1992) R1005 [hep-th/9111056].
  * (31) J.G. Russo, L. Susskind and L. Thorlacius, _The Endpoint of Hawking radiation_ , _Phys. Rev. D_ 46 (1992) 3444 [hep-th/9206070].
  * (32) S.M. Christensen and S.A. Fulling, _Trace anomalies and the hawking effect_ , _Phys. Rev. D_ 15 (1977) 2088.
  * (33) A.M. Polyakov, _Quantum Geometry of Bosonic Strings_ , _Phys. Lett. B_ 103 (1981) 207.
  * (34) W. Kummer and D.V. Vassilevich, _Effective action and hawking radiation for dilaton coupled scalars in two dimensions_ , _Physical Review D_ 60 (1999) .
  * (35) W. Kummer and D. Vassilevich, _Hawking radiation from dilaton gravity in 1 \+ 1 dimensions: a pedagogical review_ , _Annalen der Physik_ 8 (1999) 801–827.
  * (36) V. Mukhanov, A. Wipf and A. Zelnikov, _On 4d hawking radiation from the effective action_ , _Physics Letters B_ 332 (1994) 283–291.
  * (37) M. Buric, V. Radovanovic and A.R. Mikovic, _One loop correction for Schwarzschild black hole via 2-D dilaton gravity_ , _Phys. Rev. D_ 59 (1999) 084002 [gr-qc/9804083].
  * (38) R. Balbinot and A. Fabbri, _Hawking radiation by effective two-dimensional theories_ , _Physical Review D_ 59 (1999) .
  * (39) R. Balbinot and A. Fabbri, _4d quantum black hole physics from 2d models?_ , _Physics Letters B_ 459 (1999) 112.
  * (40) M. Buric and V. Radovanovic, _Quantum corrections for anti-evaporating black hole_ , _Phys. Rev. D_ 63 (2001) 044020 [hep-th/0007172].
  * (41) F.C. Lombardo, F.D. Mazzitelli and J.G. Russo, _Energy momentum tensor for scalar fields coupled to the dilaton in two-dimensions_ , _Phys. Rev. D_ 59 (1999) 064007 [gr-qc/9808048].
  * (42) Y.V. Gusev and A.I. Zelnikov, _Two-dimensional effective action for matter fields coupled to the dilaton_ , _Phys. Rev. D_ 61 (2000) 084010 [hep-th/9910198].
  * (43) D. Hofmann and W. Kummer, _Effective action and Hawking flux from covariant perturbation theory_ , _Eur. Phys. J. C_ 40 (2005) 275 [gr-qc/0408088].
  * (44) D. Hofmann and W. Kummer, _IR renormalisation of general effective actions and Hawking flux in 2-D gravity theories_ , _Eur. Phys. J. C_ 48 (2006) 291 [gr-qc/0512163].
  * (45) R. Balbinot, A. Fabbri, P. Nicolini, V. Frolov, P. Sutton and A. Zelnikov, _Vacuum polarization in the schwarzschild spacetime and dimensional reduction_ , _Physical Review D_ 63 (2001) .
  * (46) R. Balbinot, A. Fabbri, P. Nicolini and P.J. Sutton, _Vacuum polarization in two-dimensional static space-times and dimensional reduction_ , _Phys. Rev. D_ 66 (2002) 024014 [hep-th/0202036].
  * (47) A. Fabbri, S. Farese and J. Navarro-Salas, _Generalized Virasoro anomaly and stress tensor for dilaton coupled theories_ , _Phys. Lett. B_ 574 (2003) 309 [hep-th/0309160].
  * (48) D.G. Boulware, _Quantum field theory in schwarzschild and rindler spaces_ , _Phys. Rev. D_ 11 (1975) 1404.
  * (49) J.B. Hartle and S.W. Hawking, _Path Integral Derivation of Black Hole Radiance_ , _Phys. Rev. D_ 13 (1976) 2188.
  * (50) W. Israel, _Thermo field dynamics of black holes_ , _Phys. Lett. A_ 57 (1976) 107.
  * (51) P. Davies, S. Fulling and W. Unruh, _Energy Momentum Tensor Near an Evaporating Black Hole_ , _Phys. Rev. D_ 13 (1976) 2720.
  * (52) W.A. Hiscock, _Models of Evaporating Black Holes_ , _Phys. Rev. D_ 23 (1981) 2813.
  * (53) W.A. Hiscock, _Models of Evaporating Black Holes. II. Effects of the Outgoing Created Radiation_ , _Phys. Rev. D_ 23 (1981) 2823.
  * (54) A. Fabbri and A. J. Navarro-Salas, _Modeling Black Hole Evaporation_ , _Imperial College Press and World Scientific_ (2005) .
  * (55) W.G. Unruh, _Notes on black hole evaporation_ , _Phys. Rev. D_ 14 (1976) 870.
  * (56) D. Grumiller, W. Kummer and D.V. Vassilevich, _Dilaton gravity in two-dimensions_ , _Phys. Rept._ 369 (2002) 327 [hep-th/0204253].
  * (57) R. Bousso and S. Hawking, _Trace anomaly of dilaton-coupled scalars in two dimensions_ , _Phys. Rev. D_ 56 (1997) 7788.
  * (58) A. Mikovic and V. Radovanovic, _One-loop effective action for spherical scalar field collapse_ , _Classical and Quantum Gravity_ 15 (1998) 827–838.
  * (59) E. Elizalde, S. Naftulin and S.D. Odintsov, _Covariant effective action and one-loop renormalization of two-dimensional dilaton gravity with fermionic matter_ , _Phys. Rev. D_ 49 (1994) 2852.
  * (60) S. Ichinose, _Weyl anomaly of 2d dilaton-scalar gravity and the hermiticity of the system operator_ , _Phys. Rev. D_ 57 (1998) 6224.
  * (61) J.S. Dowker, _Conformal anomaly in two-dimensional dilaton-scalar theory_ , _Classical and Quantum Gravity_ 15 (1998) 1881–1884.
  * (62) M.O. Katanaev, W. Kummer, H. Liebl and D.V. Vassilevich, _Generalized 2d-dilaton models, the true black hole and quantum integrability_ , 1997.
  * (63) S. Nojiri, O. Obregon and S. Odintsov, _Unified approach to study quantum properties of primordial black holes, wormholes and of quantum cosmology_ , _Mod. Phys. Lett. A_ 14 (1999) 1309 [gr-qc/9907008].
  * (64) M. Burić and V. Radovanović, _Quantum corrections for an (anti)evaporating black hole_ , _Physical Review D_ 63 (2001) .
  * (65) R.M. Wald, _Trace anomaly of a conformally invariant quantum field in curved spacetime_ , _Phys. Rev. D_ 17 (1978) 1477.
  * (66) B.S. DeWitt, _Dynamical theory of groups and fields_ , _Conf. Proc. C_ 630701 (1964) 585.
  * (67) I.L. Shapiro, _Effective Action of Vacuum: Semiclassical Approach_ , _Class. Quant. Grav._ 25 (2008) 103001 [0801.0216].
  * (68) L. Parker and D. Toms, _Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (2009), 10.1017/CBO9780511813924.
  * (69) D.V. Vassilevich, _Heat kernel expansion: User’s manual_ , _Phys. Rept._ 388 (2003) 279 [hep-th/0306138].
  * (70) A.O. Barvinsky and G.A. Vilkovisky, _Beyond the Schwinger-Dewitt Technique: Converting Loops Into Trees and In-In Currents_ , _Nucl. Phys. B_ 282 (1987) 163.
  * (71) A.O. Barvinsky and G.A. Vilkovisky, _Covariant perturbation theory. 2: Second order in the curvature. General algorithms_ , _Nucl. Phys. B_ 333 (1990) 471.
  * (72) A.O. Barvinsky and G.A. Vilkovisky, _Covariant perturbation theory. 3: Spectral representations of the third order form-factors_ , _Nucl. Phys. B_ 333 (1990) 512.
  * (73) A.O. Barvinsky, Y.V. Gusev, V.V. Zhytnikov and G.A. Vilkovisky, _Covariant perturbation theory. 4. Third order in the curvature_ , 0911.1168.
  * (74) A.O. Barvinsky, Y.V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, _Asymptotic behaviors of the heat kernel in covariant perturbation theory_ , _J. Math. Phys._ 35 (1994) 3543 [gr-qc/9404063].
  * (75) A.O. Barvinsky and V.F. Mukhanov, _New nonlocal effective action_ , _Phys. Rev. D_ 66 (2002) 065007 [hep-th/0203132].
  * (76) A.O. Barvinsky, Y.V. Gusev, V.F. Mukhanov and D.V. Nesterov, _Nonperturbative late time asymptotics for heat kernel in gravity theory_ , _Phys. Rev. D_ 68 (2003) 105003 [hep-th/0306052].
  * (77) A.O. Barvinsky and D.V. Nesterov, _Nonperturbative heat kernel and nonlocal effective action_ , [hep-th/0402043].
  * (78) V. Frolov, P. Sutton and A. Zelnikov, _Dimensional-reduction anomaly_ , _Physical Review D_ 61 (1999) .
  * (79) G. Cognola and S. Zerbini, _On the dimensional reduced theories_ , in _Conference on Geometrical Aspects of Quantum Fields_ , pp. 64–72, 8, 2000, DOI [hep-th/0008137].
  * (80) G. Cognola and S. Zerbini, _On the dimensional reduction procedure_ , _Nucl. Phys. B_ 602 (2001) 383 [hep-th/0008061].
  * (81) D.R. Karakhanian, R.P. Manvelyan and R.L. Mkrtchian, _Area preserving structure of 2-d gravity_ , _Phys. Lett. B_ 329 (1994) 185 [hep-th/9401031].
  * (82) R. Jackiw, _Another view on massless matter - gravity fields in two-dimensions_ , hep-th/9501016.
  * (83) J. Navarro-Salas, M. Navarro and C.F. Talavera, _Weyl invariance and black hole evaporation_ , _Phys. Lett. B_ 356 (1995) 217 [hep-th/9505139].
  * (84) J.M. Bardeen, _Trace anomaly effective actions - a critique_ , 1808.09629.
  * (85) S. Bose, L. Parker and Y. Peleg, _Semiinfinite throat as the end state geometry of two-dimensional black hole evaporation_ , _Phys. Rev. D_ 52 (1995) 3512 [hep-th/9502098].
  * (86) S. Bose, L. Parker and Y. Peleg, _Hawking radiation and unitary evolution_ , _Phys. Rev. Lett._ 76 (1996) 861 [gr-qc/9508027].
  * (87) A. Fabbri and J.G. Russo, _Soluble models in 2-d dilaton gravity_ , _Phys. Rev. D_ 53 (1996) 6995 [hep-th/9510109].
  * (88) J. Cruz and J. Navarro-Salas, _Solvable models for radiating black holes and area preserving diffeomorphisms_ , _Phys. Lett. B_ 375 (1996) 47 [hep-th/9512187].
  * (89) O.B. Zaslavsky, _Exactly solvable models of two-dimensional dilaton gravity and quantum eternal black holes_ , _Phys. Rev. D_ 59 (1999) 084013 [hep-th/9804089].
  * (90) N. Engelhardt and A.C. Wall, _Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime_ , _JHEP_ 01 (2015) 073 [1408.3203].
  * (91) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _The entropy of Hawking radiation_ , _Rev. Mod. Phys._ 93 (2021) 035002 [2006.06872].
  * (92) T.M. Fiola, J. Preskill, A. Strominger and S.P. Trivedi, _Black hole thermodynamics and information loss in two-dimensions_ , _Phys. Rev. D_ 50 (1994) 3987 [hep-th/9403137].
  * (93) S. Hawking, _Particle Creation by Black Holes_ , _Commun.Math.Phys._ 43 (1975) 199.
  * (94) A. Almheiri, R. Mahajan and J. Maldacena, _Islands outside the horizon_ , 1910.11077.
  * (95) P.-M. Ho and Y. Matsuo, _Static Black Holes With Back Reaction From Vacuum Energy_ , _Class. Quant. Grav._ 35 (2018) 065012 [1703.08662].
  * (96) H. Casini and M. Huerta, _Entanglement and alpha entropies for a massive scalar field in two dimensions_ , _J. Stat. Mech._ 0512 (2005) P12012 [cond-mat/0511014].
  * (97) H. Casini and M. Huerta, _Entanglement entropy in free quantum field theory_ , _J. Phys. A_ 42 (2009) 504007 [0905.2562].
  * (98) C. Holzhey, F. Larsen and F. Wilczek, _Geometric and renormalized entropy in conformal field theory_ , _Nucl. Phys. B_ 424 (1994) 443 [hep-th/9403108].
  * (99) P. Calabrese and J. Cardy, _Entanglement entropy and conformal field theory_ , _J. Phys._ A42 (2009) 504005 [0905.4013].
  * (100) A. Almheiri and J. Polchinski, _Models of AdS 2 backreaction and holography_, _JHEP_ 11 (2015) 014 [1402.6334].
  * (101) J. Maldacena, D. Stanford and Z. Yang, _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ , _PTEP_ 2016 (2016) 12C104 [1606.01857].
  * (102) J. Engelsöy, T.G. Mertens and H. Verlinde, _An investigation of AdS 2 backreaction and holography_, _JHEP_ 07 (2016) 139 [1606.03438].
  * (103) T.G. Mertens and G.J. Turiaci, _Solvable Models of Quantum Black Holes: A Review on Jackiw-Teitelboim Gravity_ , 2210.10846.
  * (104) P. Saad, S.H. Shenker and D. Stanford, _JT gravity as a matrix integral_ , 1903.11115.
  * (105) D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, _JT gravity with matter, generalized ETH, and Random Matrices_ , 2209.02131.
  * (106) D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, _Matrix models for eigenstate thermalization_ , 2209.02130.
  * (107) S.M. Christensen and S.A. Fulling, _Trace anomalies and the hawking effect_ , _Phys. Rev. D_ 15 (1977) 2088.
  * (108) P.-M. Ho, H. Kawai, Y. Matsuo and Y. Yokokura, _Back Reaction of 4D Conformal Fields on Static Geometry_ , _JHEP_ 11 (2018) 056 [1807.11352].
  * (109) P.-M. Ho, Y. Matsuo and Y. Yokokura, _Analytic description of semiclassical black-hole geometry_ , _Phys. Rev. D_ 102 (2020) 024090 [1912.12855].