# Hawking Evaporation of Einstein-Gauss-Bonnet AdS Black Holes in $D\geqslant
4$ dimensions

Chen-Hao Wu${}^{1,}{}^{2,}{}^{4}$, Ya-Peng Hu ${}^{1,}{}^{4}$, Hao
Xu${}^{2,}{}^{3}$
1College of Science, Nanjing University of Aeronautics and Astronautics,
Nanjing, 210016, China
2Center for Gravitation and Cosmology, College of Physical Science and
Technology,
Yangzhou University, Yangzhou, 225009, China
3School of Aeronautics and Astronautics, Shanghai Jiao Tong University,
Shanghai, 200240, China
4MIIT Key Laboratory of Aerospace Information Materials and Physics,
Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
E-mail:chenhao_wu<EMAIL_ADDRESS>author,
<EMAIL_ADDRESS>
###### Abstract

Einstein-Gauss-Bonnet theory is a string-generated gravity theory when
approaching the low energy limit. By introducing the higher order curvature
terms, this theory is supposed to help to solve the black hole singularity
problem. In this work, we investigate the evaporation of the static
spherically symmetric neutral AdS black holes in Einstein-Gauss-Bonnet gravity
in various spacetime dimensions with both positive and negative couping
constant $\alpha$. By summarizing the asymptotic behavior of the evaporation
process, we find the lifetime of the black holes is dimensional dependent. For
$\alpha>0$, in $D\geqslant 6$ cases, the black holes will be completely
evaporated in a finite time, which resembles the Schwarzschild-AdS case in
Einstein gravity. While in $D=4,5$ cases, the black hole lifetime is always
infinite, which means the black hole becomes a remnant in the late time.
Remarkably, the cases of $\alpha>0,D=4,5$ will solve the terminal temperature
divergent problem of the Schwarzschild-AdS case. For $\alpha<0$, in all
dimensions, the black hole will always spend a finite time to a minimal mass
corresponding to the smallest horizon radius $r_{min}=\sqrt{2|\alpha|}$ which
coincide with an additional singularity. This implies that there may exist
constraint conditions to the choice of coupling constant.

##  1 Introduction

In 1974, by applying quantum field theory in curved spacetime, Hawking showed
that quantum mechanical effects allow black holes to emit particles, now known
as Hawking radiation [1, 2]. Hawking radiation connects quantum physics and
gravity, and it is viewed as a window to help us to understand the quantum
gravity. Due to the radiation, the black hole will keep losing mass and
thermal entropy, which leaves us the controversial information loss paradox
[3, 4].

The spectrum of the Hawking radiation allows us to calculate the particles
emission power and to estimate the lifetime of the black hole. For a static
spherically symmetric black hole in four-dimensional asymptotically flat
spacetime in Einstein gravity, its lifetime obeys the relationship $t\sim
M_{0}^{3}$ associated with the initial black hole mass $M_{0}$ [5], which
means the black hole lifetime will be divergent if we take an infinite initial
mass. A similar analysis can also be carried out in rotating black holes and
charged black holes. A rotating black hole will lose its angular momentum much
faster than its mass and eventually it becomes a Schwarzschild black hole and
keeps going on its evaporation[6](unless the case with a lot of scalar
particle species[7, 8]). A charged black hole will keep losing its charge and
mass, however, the charge-over-mass ratio may increase and gets close to the
extreme black hole, so the lifetime of these black holes may be extended by a
huge factor, but finally, it also evolves to a Schwarzschild state[9, 10].

The spectrum of the Hawking radiation allows us to calculate the particles
emission power and to estimate the lifetime of the black hole. For a static
spherically symmetric black hole in four-dimensional asymptotically flat Based
on Hawking radiation, the study on the evaporation of the black hole in
asymptotically flat spacetime has been a fruitful area. The evaporation in
anti-de Sitter (AdS) spacetimes was, however, largely overlooked. This is due
to the specific asymptotical behavior of the AdS spacetime. Although AdS has
an infinite volume, the gravitational potential acts effectively like a finite
confining box for the black hole, and the emitted massive particles reflect
inward in a finite time thus preventing the energy loss. Thanks to the AdS/CFT
correspondence [11, 13, 12, 14], a great mass of attention has been paid on
black holes in AdS spacetimes, especially AdS black hole thermodynamics[15],
and the evaporation problem of AdS black hole also comes back to our view in
recent years. By imposing an absorbing AdS boundary condition [16, 17, 18, 19]
and considering only massless emitted particle, Page calculated the lifetime
of static spherically symmetric AdS black hole in Einstein gravity [20], and
surprisingly, the lifetime did not become divergent with any arbitrarily large
initial mass and the total time to evaporate from infinite mass to zero was
finite, of the order $\ell^{3}$, where $\ell$ is the AdS radius. This is
because although the black hole may have infinite initial mass, the
temperature, and also the particles’ emission power, are also divergent, thus
the infinite amount of mass can be evaporated away in a finite time. The study
on black holes evaporation in asymptotically AdS spacetimes have been extended
to many different backgrounds, such as Lovelock gravity[21], Conformal (Weyl)
Gravity[22], Hořava-Lifshitz gravity [23], and dRGT massive gravity[24], etc.

From a modern perspective, Einstein gravity could just be an effective field
theory in the low energy limit of unknown more fundamental theory. Although we
have not known its correct and precise form, there are some attempts by
physicists. Inspired by the quadratic term in heterotic string theory and six-
dimensional Calabi-Yau compactifications of M-theory, the Gauss-Bonnet
quadratic term was added into the Einstein-Hilbert action to analyze the
properties of the effective gravity theory[25, 27, 28, 26, 29]. This gravity
can be derived by string theories when approaching the low energy limit.
Studies on the Gauss-Bonnet gravity and its generalizations have attracted
lots of attention [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], however, this
theory just exists in higher spacetime dimensions because the Gauss-Bonnet
term is a topological invariant in four dimensions thus it cannot give any
non-trivial gravitational dynamics in $D\leq 4$, where $D$ is the dimensions
of the spacetime. In [41], Glavan and Lin try to extend this theory to four
dimensions by rescaling the coupling constant of the Gauss-Bonnet term by a
factor of $1/(D-4)$ and taking the limit $D\rightarrow 4$. Remarkably, from
their point of view, this theory can bypass the Lovelock’s theorem and be free
from Ostrogradsky instability. Interestingly, its black hole solution was also
found in other contexts, such as gravity with conformal anomaly[42, 43] and
quantum corrections[44]. This processing method in [41] has led to a great
deal of discussion, see e.g. [45, 46, 47, 48, 49, 53, 54, 55, 50, 51, 52, 56,
57, 58, 59, 60].

In the present work, we investigate the Hawking evaporation process of the
static spherically symmetric neutral AdS black hols in Einstein-Gauss-Bonnet
gravity. The thermodynamical properties of these black holes in $D\geqslant 5$
and $D=4$ are provided in [30, 46]. Unlike the Schwarzschild-AdS case, which
always has a divergent temperature in the late period of the evaporation
process[61], the Einstein-Gauss-Bonnet theory reveals richer phenomena by
choosing different coupling constant. By testing the Hawking evaporation
processes of different Einstein-Gauss-Bonnet AdS black holes, we give some
clues to constrain the choice of coupling constant. Especially, the extreme
behavior of the cases with negative coupling constant also resembles the
Garfinkle-Horowitz-Strominger dilation black hole[62].

This paper is organized as follows: In Sec.2, we briefly review the
thermodynamics of $D\geqslant 5$-dimensional neutral Einstein-Gauss-Bonnet AdS
black holes. By investigating the Hawking evaporation process of the $D=5$ and
$D\geqslant 6$ cases, we show that the positive coupling constant may help to
solve divergent temperature, while the negative one may make this problem
worse. In Sec.3, our investigation is extended to the $D=4$ cases. Summary
remarks are given in Sec.4. We adopt the natural unit system, setting the
speed of light in vacuum $c$, the gravitational constant $G_{N}$, the Planck
constant $h$, and the Boltzmann constant $k_{B}$ equal to one.

##  2 Evaporation of Einstein-Gauss-Bonnet AdS Black Holes in $D\geqslant 5$

In this section, first, we will give a brief review of Einstein-Gauss-Bonnet
AdS black holes for $D\geqslant 5$ dimensions, then we investigate the black
hole evaporation process.

###  2.1 Black Hole Thermodynamics in $D\geqslant 5$ Einstein-Gauss-Bonnet AdS
spacetimes

For the $D\geqslant 5$ dimensional AdS spacetime, the Einstein-Hilbert action
with a Gauss-Bonnet term can be wriiten as

$S=\frac{1}{16\pi}\int
d^{D}x\sqrt{-g}\left(\mathcal{R}+\frac{(D-1)(D-2)}{\ell^{2}}+\alpha_{\text{GB}}\mathcal{L}_{\text{GB}}\right),$
(1)

where $\alpha_{\text{GB}}$ is the Gauss-Bonnet coupling coefficient with
dimension $(length)^{2}$ and the Gauss-Bonnet term
$\mathcal{L}_{\text{GB}}=\mathcal{R}_{\mu\nu\gamma\delta}\mathcal{R}^{\mu\nu\gamma\delta}-4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}+\mathcal{R}^{2}$.
The Gauss-Bonnet coefficient $\alpha_{\text{GB}}$ is supposed to be a postive
value in string theory[25], but for generality we consider both cases of
$\alpha_{\text{GB}}>0$ and $\alpha_{\text{GB}}<0$ in the present work. The
static spherically symmetric black hole metric in $D$ dimensions can be
written in the form of

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega_{D-2}^{2},$ (2)

with the the metric function[30]

$f(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{64\pi
M\alpha}{(D-2)\Omega_{D-2}r^{D-1}}-\frac{4\alpha}{\ell^{2}}}\right),$ (3)

where $\alpha=(D-3)(D-4)\alpha_{\text{GB}}$ and the parameter $M$ is the
gravitational mass of the solution. The
$\Omega_{D-2}=\frac{(D-1)\pi^{D/2}}{\Gamma(\frac{D+1}{2})}$ is the area of an
$(D-1)$-dimensional sphere with unit radius. When $\alpha\rightarrow 0$, black
hole solution reduces to the Schwarzschild AdS case in [63]. In the limit
$M\rightarrow 0$, one can obtain the vacuum solution
$f(r)=1+r^{2}/\ell^{2}_{eff}$ with the effective AdS radius
$\ell_{eff}^{2}=\frac{\ell^{2}}{2}\left(1+\sqrt{1-\frac{4\alpha}{\ell^{2}}}\right)$,
which implies the coupling constant $\alpha$ must obey
$\alpha\leqslant\ell^{2}/4$, beyond that this theory is undefined. The event
horizon of the black hole is determined by $f(r_{+})=0$, and the mass of black
hole $M$ can be obtained as [30]

$M=\frac{(D-2)\Omega_{D-2}r_{+}^{D-3}}{16\pi}\left(1+\frac{\alpha}{r_{+}^{2}}+\frac{r_{+}^{2}}{l^{2}}\right),$
(4)

and the Hawking temperature of the black hole can be written as

$T=\left.\frac{f^{\prime}(r)}{4\pi}\right|_{r=r_{+}}=\frac{(D-1)r_{+}^{4}+(D-3)\ell^{2}r_{+}^{2}+(D-5)\alpha\ell^{2}}{4\pi\ell^{2}r_{+}(r_{+}^{2}+2\alpha)}.$
(5)

In $\alpha>0$, it is worth noting that there is a mass gap $M_{c}\equiv
M(r_{+}\rightarrow 0)=3\pi\alpha/8$ in $D=5$ that all black holes must satisfy
$M\geqslant M_{c}$, while in $D\geqslant 6$, there exists no mass gap. We can
also find the last term in (5) vanishes when $D=5$. Consequently, the
temperature $T$ vanishes as $r_{+}\rightarrow 0$ when $D=5$, while it becomes
divergent as $r_{+}\rightarrow 0$ in higher dimensions. Among various choices
for the spacetime dimension $D$, the particular case $D=5$ is qualitatively
different from other choices. This is perhaps a consequence of $D=5$ is the
lowest dimension in which the Gauss-Bonnet term can affect the local geometry.
In $\alpha<0$, there is a extreme horizon radius $r_{min}=\sqrt{2|\alpha|}$
corresponding to a divergent temperature, below that the theory is undefined.
The black hole entropy can also be calculated by using the first law of black
hole thermodynamics, as well as other thermodynamical quantities. We omit them
here.

###  2.2 Black Hole Evaporation in $D\geqslant 5$ Einstein-Gauss-Bonnet AdS
spacetimes

Since we have obtained all the necessary thermodynamical quantities, we are
ready to investigate the evaporation of Einstein-Gauss-Bonnet AdS black holes.
By imposing an absorbing boundary condition, the black hole mass $M$ should be
monotonically-decreasing functions of the time $t$ because of the Hawking
radiation. Applying geometric optics approximation, we assume all the emitted
massless particles move along null geodesics. If we orient the extra $(D-3)$
angular coordinates in $\mathrm{d}\Omega_{D-2}^{2}$ and normalize the affine
parameter $\lambda$, the geodesic equation of the massless particles reads

$\displaystyle\bigg{(}\frac{\mathrm{d}r}{\mathrm{d}\lambda}\bigg{)}^{2}=E^{2}-J^{2}\frac{f(r)}{r^{2}},$
(6)

where $E=f(r)\frac{\mathrm{d}t}{\mathrm{d}\lambda}$ is the energy and
$J=r^{2}\frac{\mathrm{d}\theta}{\mathrm{d}\lambda}$ is angular momentum of the
emitted particle. Consider an emitted particle coming from just outside the
black hole horizon, this particle cannot be detected by the observer on the
AdS boundary if there is a turning point satisfying
$\big{(}\frac{\mathrm{d}r}{\mathrm{d}\lambda}\big{)}^{2}=0$. Defining an
impact parameter $b\equiv{J}/{E}$, the massless particle can reach infinity
only if

$\displaystyle\frac{1}{b^{2}}>\frac{f(r)}{r^{2}},$ (7)

for all $r>r_{+}$. The impact factor $b_{c}$ is defined by the maximal value
of ${f(r)}/{r^{2}}$, which depends on the exact form of metric function
${f(r)}$.

If we obtain the impact factor $b_{c}$, we can get the Hawking emission rate
in $D$-dimensional spacetime according to Stefan-Boltzmann law, yields[64, 65,
10, 21, 23]

$\displaystyle\frac{\mathrm{d}M}{\mathrm{d}t}=-Cb_{c}^{D-2}T^{D},$ (8)

with the constant
$C=(D-2)\pi^{\frac{D}{2}-1}\Omega_{D-2}\frac{k^{D}}{h^{D-1}c^{D-1}}\frac{\Gamma(D)}{\Gamma(\frac{D}{2})}\zeta(D)$.
Without loss of generality, we set the constant $C=1$. The Stefan-Boltzmann
law implies that in $D$-dimensional spacetime the emission power is
proportional to the $(D-2)$-dimensional cross section $b_{c}^{D-2}$ and the
$(D-1)$-dimensional space (spatial dimension only) photon energy $T^{D}$.
Notice that the photon energy term $T^{D}$ has the higher order, so the
behavior of temperature $T$, especially the asymptotical behavior, will play a
leading role in black hole evaporation process. Next, we are ready to study
the black hole evaporation for various features of $T$ and $b_{c}$.

Using scaling analysis we know $M\sim l^{D-3}$, $T\sim l^{-1}$, $b_{c}\sim l$,
$\ell\sim l$ and $\alpha\sim l^{2}$, where $l$ has the dimension $(length)$.
In order to get dimensionless parts of $M$, $T$ and $b_{c}$, by defining
dimensionless variables $x\equiv{r_{+}}/{\ell}$ and
$y\equiv{\alpha}/{\ell^{2}}$, we can express $M$ and $T$ as

$M=\frac{(D-2)\Omega_{D-2}}{16\pi}\left(1+y+x^{2}\right)x^{D-3}\ell^{D-3},$
(9) $T=\frac{(D-1)x^{4}+(D-3)x^{2}+(D-5)y}{4\pi x(x^{2}+2y)}\ell^{-1}.$ (10)

The $b_{c}$ depends on the exact form of metric function $f(r)$, so we cannot
get the universal solution for various dimensions.

Firstly we start from the case of $D=5$, which is qualitatively different from
the cases of $D\geqslant 6$. When $D=5$, the temperature (5) can be written as

$T=\frac{2r_{+}^{3}+\ell^{2}r_{+}}{2\pi\ell^{2}(r_{+}^{2}+2\alpha)}.$ (11)

In FIG.1 we present the temperature of black holes in $5$ dimensions with
$\alpha>0$ and $\alpha<0$ as the function of $r_{+}$ respectively. For
$\alpha>0$, the temperature starts from zero at $r_{+}=0$ and goes to infinity
when $r_{+}\rightarrow\infty$. The behavior of temperature in $\alpha<0$ is
quite different from the case of $\alpha>0$. There is an extreme horizon
radius $r_{min}=\sqrt{2|\alpha|}$, which coincides with an additional
singularity. As shown in the figure, the temperature is divergent at $r_{min}$
when $\alpha<0$, and as a result, the thermodynamical properties of case
$\alpha<0$ resembles the case of $\alpha\rightarrow 0$ that without Gauss-
Bonnet term.

Figure 1: The temperature of black holes in $5$ dimensions with $\alpha>0$ and
$\alpha<0$ respectively. The left figure we set $\alpha=0.15$, $\ell=1$, while
in the right figure $\alpha=-0.15$, $\ell=1$.

In order to get the impact factor $b_{c}$, solving the equation
$\frac{\partial}{\partial r}\frac{f(r)}{r^{2}}=0$, we can find the root

$r_{p}=\frac{2}{\sqrt{3\pi}}\left(\frac{6\pi\ell^{2}M\alpha-16\ell^{2}M^{2}}{4\alpha-\ell^{2}}\right)^{1/4}.$
(12)

This radius corresponds to the unstable photon orbit, and the impact factor is
given by $b_{c}={r_{p}}/{\sqrt{f(r_{p})}}$. Similarly, taking
$x\equiv{r_{+}}/{\ell}$ and $y\equiv{\alpha}/{\ell^{2}}$, one can obtain the
$b_{c}$ as

$b_{c}=\left(\frac{\sqrt{4y-1}}{2\sqrt{-(x^{2}+x^{4})(x^{2}+x^{4}+y)}}-\frac{-1+\sqrt{-\frac{(x^{2}+x^{4}+y)(4y-1)}{x^{2}+x^{4}}}}{2y}\right)^{-\frac{1}{2}}\ell$
(13)

For $D=5$, inserting the black hole mass $M$ (9), temperature $T$ (10) and
impact factor $b_{c}$ (13) into Stefan-Boltzmann law, we can obtain

$\displaystyle\mathrm{d}t=\ell^{4}F_{1}(x,y)\mathrm{d}x,$ (14)

where $F_{1}(x,y)$ is a complicated function which is not worth explicitly
written. Setting $y$ to be a constant and integrating the equation from
$\infty$ to $x_{min}$, where $x_{min}=0$ for $\alpha>0$ and
$x_{min}=r_{min}/\ell=\sqrt{2|y|}$ for $\alpha<0$, one can obtain the black
hole lifetime is divergent in $\alpha>0$ and of the order $\ell^{4}$ in
$\alpha<0$.

In FIG.2, we plot some numerical results of the black hole mass $M$ as
function of the lifetime $t$ by setting $y>0$ and $y<0$ (corresponding to
$\alpha>0$ and $\alpha<0$ respectively). In the case of $\alpha>0$, we set
$y=0.1$, and the initial mass is taken to infinity. The black holes lose their
mass in a “short” time, but the evaporation gets harder when the black hole
mass and temperature become smaller near the $T=0$ state, thus the black hole
will have an infinite lifetime, satisfying the third law of black hole
thermodynamics. This means that the black hole becomes a remnant, which may
help us to ameliorate the information paradox [66]. In the case $\alpha<0$,
there is an extreme horizon radius at $r_{min}=\sqrt{2|\alpha|}$, where the
temperature of black hole is divergent coinciding with an additional
singularity [30]. This $r_{min}$ corresponds to a mass $M_{min}=M(r_{min})$,
and the black hole will take a finite time to decrease the mass to $M_{min}$.
Interestingly, this phenomenon resembles the extreme case of Garfinkle-
Horowitz-Strominger dilation black hole[62].

Figure 2: The numerical results of the black hole mass $M$ as function of the
lifetime $t$ with setting $y>0$ and $y<0$ (corresponding to $\alpha>0$ and
$\alpha<0$, respectively). In the left figure, we set $y=0.1$ with $\ell=1$,
$\ell=1.5$ and $\ell=2$ from left to right respectively. Note that there is a
mass gap $M_{c}=\frac{3\pi\alpha}{8}$ for $y>0$ ($\alpha>0$). In the right
figure, we set $y=-0.1$ with $\ell=1$, $\ell=1.5$ and $\ell=2$ from left to
right, corresponding to minimum mass $M_{min1}=0.164934$, $M_{min2}=0.371101$
and $M_{min3}=0.659734$, respectively.

Now we extend our study to $D\geqslant 6$. The temperature behavior in
$D\geqslant 6$ is quite different from the case $D=5$ when $\alpha>0$. For
$D\geqslant 6$, the temperature is divergent at $r_{+}=0$ and goes to infinity
again when $r_{+}\rightarrow\infty$[30], which resembles the case of
Schwarzschild AdS. On the other hand, the temperature behavior of case
$\alpha<0$ in $D\geqslant 6$ spacetimes is similarly with the case $\alpha<0$
in 5 dimensions, where there is also the smallest horizon radius at
$r_{min}=\sqrt{2|\alpha|}$. Although $\frac{\partial}{\partial
r}\frac{f(r)}{r^{2}}=0$ is a equation of higher order in $D\geqslant 6$
dimensions and we cannot get the explicit form of $b_{c}$, we can still
conclude $b_{c}\sim\ell$ by scaling analysis. As we have emphasized earlier in
the present work, the photon energy term $T^{D}$ plays the leading role in the
black hole evaporation process. Similarly, inserting the black hole mass
$M\sim\ell^{D-3}$, temperature $T\sim\ell^{-1}$ and impact factor
$b_{c}\sim\ell$ into Stefan-Boltzmann law, we can also obtain the
$D$-dimensional Stefan-Boltzmann law as

$\displaystyle\mathrm{d}t=\ell^{D-1}F_{2}(x,y)\mathrm{d}x,$ (15)

where $F_{2}(x,y)$ is still a complicated function. We can conclude this
evaporation process qualitatively by analyzing the temperature asymptotic
behavior of the black hole. When $\alpha>0$, the temperature $T$ is divergent
in both $r_{+}\rightarrow\infty$ and $r_{+}\rightarrow 0$, which means that
$\int^{0}_{\infty}F_{2}(x,y)\mathrm{d}x$ is convergent and just depends on the
value of $y$. Hence, the lifetime of the black hole will be a finite value of
the order $\ell^{D-1}$ with any arbitrarily large initial mass, which
resembles the Schwarzschild AdS black hole in Einstein gravity [20]. For
$\alpha<0$, the temperature of black hole is still divergent at
$r_{min}=\sqrt{2|\alpha|}$, thus the qualitative features of the evaporation
process in $D\geqslant 6$ dimensions are consistent with the case of
$\alpha<0$ in $5$ dimensions.

##  3 Evaporation of novel $4D$ Einstein-Gauss-Bonnet AdS Black Holes

Since Gauss-Bonnet term is a topological invariant in four dimensions,
previously the investigation of Gauss-Bonnet gravity is limited to $D\geq 5$
dimensions. This situation is changed when Glavan & Lin declare that rescaling
the coupling coefficient $\alpha_{\text{GB}}$ of the Gauss-Bonnet term by a
factor of $1/(D-4)$ can lead to non-trivial dynamics when taking the limit
$D\rightarrow 4$, thus extend the classic Einstein-Gauss-Bonnet theory to $4$
dimensions [41]. Remarkably, in their point of view, this theory can bypass
Lovelock’s theorem and be free from Ostrogradsky instability. The black hole
solution in AdS space has been provided in [46]. It is worth noting that this
black hole solution has been found in other backgrounds, such as gravity with
conformal anomaly [42, 43] and with quantum corrections[44]. In this section,
we will take the metric in [46] and investigate the qualitative evaporation
process. We hope our result may help inspiring the researches on $4D$
Einstein-Gauss-Bonnet gravity.

###  3.1 Black Hole Thermodynamics in $4D$ Einstein-Gauss-Bonnet AdS
spacetimes

In this section we give a brief review on thermodynamics of $4D$ Einstein-
Gauss-Bonnet AdS black holes. The static spherically symmetric balck hole
solution takes the form

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega_{2}^{2}$ (16)

with the the metric function[46]

$f(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+4\alpha\left(\frac{2M}{r^{3}}-\frac{1}{\ell^{2}}\right)}\right),$
(17)

where $\alpha$ is coupling constant and the parameter $M$ is the mass of black
hole. Similarly, in the vanishing limit of mass $M\rightarrow 0$, one can
obtain the vacuum solution $f(r)=1+r^{2}/\ell^{2}_{eff}$ with the effective
AdS radius
$\ell_{eff}^{2}=\frac{\ell^{2}}{2}\left(1+\sqrt{1-\frac{4\alpha}{\ell^{2}}}\right)$,
thus the couping constant $\alpha$ must obey $\alpha\leqslant\ell^{2}/4$. The
solution still keeps asymptotically AdS when $\alpha<0$. Notably, the black
hole solution will recover the solution in Einstein gravity when
$\alpha\rightarrow 0$. Solving $f(r_{+})=0$ at the event horizon $r_{+}$, we
can obtain the mass of black hole

$M=\frac{r_{+}}{2}\left(1+\frac{r_{+}^{2}}{\ell^{2}}+\frac{\alpha}{r_{+}^{2}}\right).$
(18)

The Hawking temperature of the black hole can be obtain as

$T=\left.\frac{f^{\prime}(r)}{4\pi}\right|_{r=r_{+}}=\frac{3r_{+}^{4}+\ell^{2}(r_{+}^{2}-\alpha)}{4\pi\ell^{2}r_{+}(r_{+}^{2}+2\alpha)},$
(19)

For $\alpha>0$, solving $T=0$, there is a critical horizon radius
$r_{c}=\sqrt{\left(-\ell^{2}+\sqrt{\ell^{2}(\ell^{2}+12\alpha)}\right)/6}$
corresponding to a vanishing temperature. Similarly there will also be a
critical mass

$M_{c}=\frac{-\ell^{2}+12\alpha+\sqrt{\ell^{4}+12\ell^{2}\alpha}}{3\sqrt{6}\sqrt{-\ell^{2}+\sqrt{\ell^{4}+12\ell^{2}\alpha}}},$
(20)

which depends on the AdS radius $\ell$ and the coupling constant $\alpha$. For
$M<M_{c}$, there is no black hole. The black hole entropy can also be
calculated by the first law of black hole thermodynamics, see [46].

###  3.2 Black Hole Evaporation in $4D$ Einstein-Gauss-Bonnet AdS spacetimes

After the review of the thermodynamics of $4D$ Einstein-Gauss-Bonnet AdS black
holes, we are ready to explore the black holes evaporation. Imposing the same
assumptions that we take in the case $D\geqslant 5$, we can obatin the Stefan-
Boltzmann law in $4D$, reads

$\displaystyle\frac{\mathrm{d}M}{\mathrm{d}t}=-b_{c}^{2}T^{4}.$ (21)

Note that the behavior of temperature $T$ still plays a leading role in black
hole evaporation process since it occupies the highest order. In order to
analyze the qualitative behavior of the black hole temperature $T$, we provide
the examples with $\alpha>0$ and $\alpha<0$ in FIG.3. It is clear that $T=0$
at a critical horizon radius
$r_{c}=\sqrt{(-\ell^{2}+\sqrt{\ell^{2}(\ell^{2}+12\alpha)})/6}$, which means
that exists a critical mass $M_{c}$. When $\alpha<0$, this situation resembles
the case $\alpha<0$ in $D\geqslant 5$ which also have divergent temperature at
$r_{min}=\sqrt{2|\alpha|}$.

Figure 3: The temperature of black holes in $4$-dimensions with $\alpha>0$ and
$\alpha<0$ respectively. The left figure we set $\alpha=0.15$, $\ell=1$, while
in the right $\alpha=-0.15$, $\ell=1$.

Solving the equation $\frac{\partial}{\partial r}\frac{f(r)}{r^{2}}=0$, we can
obtain the root corresponding to the photon orbit as

$\displaystyle
r_{p}=\left(-\frac{4M\alpha}{1-\frac{4\alpha}{\ell^{2}}}-\sqrt{-\frac{27M^{6}}{(1-\frac{4\alpha}{\ell^{2}})^{3}}+\frac{16M^{2}\alpha^{2}}{(1-\frac{4\alpha}{\ell^{2}})^{2}}}\right)^{\frac{1}{3}}+\left(-\frac{4M\alpha}{1-\frac{4\alpha}{\ell^{2}}}+\sqrt{-\frac{27M^{6}}{(1-\frac{4\alpha}{\ell^{2}})^{3}}+\frac{16M^{2}\alpha^{2}}{(1-\frac{4\alpha}{\ell^{2}})^{2}}}\right)^{\frac{1}{3}}.$
(22)

Similarly, using the dimensionless variables $x\equiv{r_{+}}/{\ell}$ and
$y\equiv{\alpha}/{\ell^{2}}$, we can re-express the black hole mass (18) and
the temperature (19) as

$\displaystyle M$ $\displaystyle=\frac{x^{2}+x^{4}+y}{2x}\ell,$ (23)
$\displaystyle T$ $\displaystyle=\frac{x^{2}+3x^{4}-y}{4\pi x^{3}+8\pi
xy}\ell^{-1}.$ (24)

The impact factor $b_{c}={r_{p}}/{\sqrt{f(r_{p})}}$ can also be expressed in
the form of

$\displaystyle b_{c}$
$\displaystyle=\left(\frac{4}{\left(2(A-B)^{1/3}+2(A+B)^{1/3}\right)^{2}}-\frac{-1+\sqrt{1-4y+\frac{32y(x^{2}+x^{4}+y)}{x\left(2(A-B)^{1/3}+2(A+B)^{1/3}\right)^{3}}}}{2y}\right)^{-\frac{1}{2}}\ell$
$\displaystyle A$ $\displaystyle=\frac{2y(x^{2}+x^{4}+y)}{x(-1+4y)},\quad
B=\frac{1}{8}\sqrt{\frac{(x^{2}+x^{4}+y)^{2}(256x^{4}(1-4y)y^{2}-27(x^{2}+x^{4}+y)^{4})}{x^{6}(1-4y)^{3}}}.$
(25)

Inserting the above $M$, $T$ and $b_{c}$ into the Stefan-Boltzmann law in $4D$
spacetimes, we can obtain the lifetime relationship of black hole evaporation,
reads

$\displaystyle\mathrm{d}t=\ell^{3}F_{3}(x,y)\mathrm{d}x,$ (26)

where $F_{3}(x,y)$ is a complex function related to the $M$, $T$ and $b_{c}$.
Similarly, integrating this equation from $\infty$ to $x_{min}$ (where the
temperature $T$ has divergence or $T=0$), one can find that the lifetime is of
the order $\sim\ell^{3}$ with $\alpha<0$ or divergent with $\alpha>0$, which
resembles the case in $D\geqslant 5$ dimensions. We plot some numerical
results of the $4D$ Einstein-Gauss-Bonnet AdS black hole in FIG.4. As shown in
the figure, the case of $\alpha>0$ is very similar to the case of $5D$ with
$\alpha>0$, just has the difference in the value of critical mass which in
case $5D$ is $3\pi\alpha/8$, while in case $4D$ equals to $M_{c}$. While in
$\alpha<0$, the qualitative features of the evaporation process are parallel
to the case $\alpha<0$ in $5D$, which also possess a minimal horizon radius at
$r_{min}=\sqrt{2|\alpha|}$. The qualitative features of the evaporation
process identify what implied from the asymptotical behavior of the black hole
temperature $T$.

Figure 4: The numerical results of the $4D$ Einstein-Gauss-Bonnet AdS black
hole mass $M$ as function of the lifetime $t$ with setting $y>0$ and $y<0$
(corresponding to $\alpha>0$ and $\alpha<0$, respectively). In the left
figure, we set $y=0.1$ with $\ell=1$, $\ell=1.5$ and $\ell=2$ from left to
right, corresponding to critical mass $M_{c1}=0.329509$, $M_{c2}=0.494264$ and
$M_{c3}=0.659019$, respectively. In the right figure, we set $y=-0.1$ with
$\ell=1$, $\ell=1.5$ and $\ell=2$ from left to right, corresponding to minimum
mass $M_{min1}=0.156525$, $M_{min2}=0.234787$ and $M_{min3}=0.313050$,
respectively.

##  4 Summary

By introducing the higher order curvature term, Einstein-Gauss-Bonnet theory
is supposed to solve a class of singularity problem of black holes. Because of
the modification, the thermodynamic properties of Einstein-Gauss-Bonnet AdS
black holes are quite different from the Schwarzschild-AdS case. In the
present work, we investigate the evaporation process of Einstein-Gauss-Bonnet
AdS black holes in $D$-dimensional cases with both positive and negative
coupling constant. Firstly, for the case of $D=5$ and $\alpha>0$, the
temperature $T$ is not divergent at a small distance and admits the state of
$T=0$ at $r_{+}=0$, which is different from the Schwarzschild-AdS case.
Applying the geometric optics approximation and absorbing AdS boundary
condition, we investigate the evaporation of black holes by using the
$D$-dimensional Stefan-Boltzmann law. The black holes will lose an infinite
amount of mass in a “short” time, then the evaporation will become slower and
slower near the $T=0$ state, which means that the black holes with $\alpha>0$
have a divergent lifetime in $5$-dimensional spacetime. This phenomenon also
obeys the third law of black hole thermodynamics. When $\alpha<0$, the
temperature is divergent at $r_{min}=\sqrt{2|\alpha|}$, and the black holes
will always spend finite time $t\sim\ell^{4}$ to reach the extreme mass
$M(r_{min})$. For the cases of $D\geqslant 6$, although it is difficult to get
the exact form of impact factor $b_{c}$, we can still dope out the evaporation
process by analyzing the asymptotic behavior of the temperature. Since its
temperature is always positive and divergent in $r_{+}\rightarrow 0$ and
$r_{+}\rightarrow\infty$, the black hole lifetime is always of the order
$t\sim\ell^{D-1}$, which resembles the Schwarzschild-AdS case. Applying the
“novel” metric, we extend our investigation to $4D$. Just with a difference in
the value of critical mass between case $4D$ and case $5D$ when $\alpha>0$,
the evaporation behaviors of both $\alpha>0$ and $\alpha<0$ are parallel to
the ones in $5D$.

In the present work, we consider both $\alpha>0$ and $\alpha<0$ in different
dimensions and find that some choices of $\alpha$ are not applicable. The case
of $\alpha>0$ may help us solving the problem of the divergent temperature in
the terminal of the evaporation process of the Schwarzschild-AdS case, while
the $\alpha<0$ could make it worse. Also, in the cases $D=4,5$ there always
have remnants, while in the cases $D\geqslant 6$ that all the black holes will
evaporate in a finite time. This phenomenon is quite similar to the Hořava-
Lifshitz case[23], which also has the dimensional dependent evaporation
behavior.

Finally, we suggest some topics that can be further investigated. In
consideration of the lifetime of the black holes in Einstein-Gauss-Bonnet and
Hořava-Lifshitz gravity are dimensional dependent, there may be some
relationship between the dimensions $D$ and asymptotic behavior of
evaporation, finding this relationship may become an interesting work[67, 68].
Besides, this work can be extended to charged black holes in Einstein-Gauss-
Bonnet gravity [69, 46]. By considering the Schwinger effect and the Stefan-
Boltzmann law, we could calculate the charged particles emission and the mass
loss[10]. How to describe the evolution of charged black holes in Einstein-
Gauss-Bonnet gravity will be challenging and meaningful work. Furthermore,
modification to the AdS/CFT correspondence can be considered from the
perspective of Einstein-Gauss-Bonnet AdS black hole evaporation. In the
present work, our derivation relies on the geometric optics approximation and
absorbing boundary condition. Nowadays there have been many papers on
evaporating black hole in AdS in connection with the formation of islands [70,
71, 72]. It would be interesting to study the connection among absorbing
boundary conditions, black hole evaporation in AdS spacetime, and the islands.
We hope to consider these questions in our future work.

## Acknowledgements

We would like to thank Yen Chin Ong, Rui-Hong Yue and De-Cheng Zou for useful
discussion. This work is supported by National Natural Science Foundation of
China (NSFC) under grant No. 11575083, 11565017, and the Natural Science
Foundation of the Jiangsu Higher Education Institutions of China (Grant No.
20KJD140001).

## References

  * [1] S. W. Hawking, “Black Hole Explosions”, Nature 248 (1974) 30.
  * [2] S. W. Hawking, “Particle Creation by Black Holes”, Commun. Math. Phys. 43 (1975) 199; Erratum: Commun. Math. Phys. 46 (1976) 206.
  * [3] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, “Black Holes: Complementarity or Firewalls?”, JHEP 1302 (2013) 062 [arXiv:1207.3123 [hep-th]].
  * [4] S. W. Hawking, M. J. Perry and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes”, JHEP 1705 (2017) 161 [arXiv:1611.09175 [hep-th]].
  * [5] D. N. Page, “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole”, Phys. Rev. D 13 (1976) 198.
  * [6] D. N. Page, “Particle Emission Rates from a Black Hole. 2. Massless Particles from a Rotating Hole”, Phys. Rev. D 14 (1976) 3260.
  * [7] Chris M. Chambers, William A. Hiscock and Brett E. Taylor, “The ‘Ups’ and ‘Downs’ of a Spinning Black Hole”, [arXiv:gr-qc/9710013].
  * [8] Brett E. Taylor, Chris M. Chambers and William A. Hiscock, “Evaporation of a Kerr Black Hole by Emission of Scalar and Higher Spin Particles”, Phys. Rev. D 58 (1998) 044012, [arXiv:gr-qc/9801044].
  * [9] W. A. Hiscock and L. D. Weems, “Evolution of Charged Evaporating Black Holes”, Phys. Rev. D 41 (1990) 1142.
  * [10] H. Xu, Y. C. Ong and M. H. Yung, “Cosmic Censorship and the Evolution of d-Dimensional Charged Evaporating Black Holes”, Phys. Rev. D 101 (2020) 064015, [arXiv:1911.11990 [gr-qc]].
  * [11] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,”  Int. J. Theor. Phys. 38 (1999), 1113-1133 [arXiv:hep-th/9711200].
  * [12] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,”  Adv. Theor. Math. Phys. 2 (1998), 505-532 [hep-th/9803131].
  * [13] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998), 253-291 [arXiv:hep-th/9802150].
  * [14] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,”  Phys. Lett. B 428 (1998), 105-114 [arXiv:hep-th/9802109].
  * [15] D. Kubiznak and R. B. Mann, “Black hole chemistry,”  Can. J. Phys. 93 (2015) no.9, 999-1002 [arXiv:1404.2126 [gr-qc]].
  * [16] S. J. Avis, C. J. Isham and D. Storey, “Quantum Field Theory in Anti-de Sitter Spacetime”, Phys. Rev. D 18 (1978) 3565
  * [17] Jorge V. Rocha, “Evaporation of Large Black Holes in AdS: Coupling to the Evaporon”, JHEP 08 (2008) 075, [arXiv:0804.0055 [hep-th].]
  * [18] Ahmed Almheiri, Donald Marolf, Joseph Polchinski, Douglas Stanford, James Sully, ‘‘An Apologia for Firewalls’’, JHEP 09 (2013) 018, [arXiv:1304.6483 [hep-th].]
  * [19] Mark Van Raamsdonk, ‘‘Evaporating Firewalls’’, JHEP 11 (2014) 038, [arXiv:1307.1796 [hep-th].]
  * [20] D. N. Page, ‘‘Finite Upper Bound for the Hawking Decay Time of an Arbitrarily Large Black Hole in Anti-de Sitter Spacetime’’, Phys. Rev. D 97 (2018) 024004 [arXiv:1507.02682 [hep-th]].
  * [21] H. Xu and M. H. Yung, ‘‘Black Hole Evaporation in Lovelock Gravity With Diverse Dimensions’’, Phys. Lett. B 794 (2019) 77 [arXiv:1904.06503 [gr-qc]].
  * [22] H. Xu and M. H. Yung, ‘‘Black Hole Evaporation in Conformal (Weyl) Gravity’’, Phys. Lett. B 793 (2019) 97 [arXiv:1811.07309 [gr-qc]].
  * [23] H. Xu and Y. C. Ong, ‘‘Black hole evaporation in Ho$\check{r}$ava–Lifshitz gravity,’’  Eur. Phys. J. C 80 (2020) no.7, 679 [arXiv:2005.02086 [gr-qc]].
  * [24] M. S. Hou, H. Xu and Y. C. Ong, ‘‘Hawking Evaporation of Black Holes in Massive Gravity,’’  Eur. Phys. J. C 80 (2020) no.11, 1090 [arXiv:2008.10049 [hep-th]].
  * [25] D. G. Boulware and S. Deser, ‘‘String Generated Gravity Models,’’ Phys. Rev. Lett. 55 (1985), 2656.
  * [26] C. G. Callan, Jr., R. C. Myers and M. J. Perry, ‘‘Black Holes in String Theory,’’ Nucl. Phys. B 311 (1989), 673-698.
  * [27] D. Lovelock, ‘‘The Einstein Tensor and Its Generalizations’’, J. Math. Phys. 12 (1971) 498.
  * [28] C. Garraffo and G. Giribet, ‘‘The Lovelock Black Holes,’’  Mod. Phys. Lett. A 23 (2008), 1801-1818 [arXiv:0805.3575 [gr-qc]].
  * [29] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, ‘‘Modified Gravity and Cosmology,’’ Phys. Rept. 513 (2012), 1-189. [arXiv:1106.2476 [astro-ph.CO]].
  * [30] R. G. Cai, ‘‘Gauss-Bonnet black holes in AdS spaces,’’ Phys. Rev. D 65 (2002), 084014. [arXiv:hep-th/0109133].
  * [31] M. Cvetic, S. Nojiri and S. D. Odintsov, ‘‘Black hole thermodynamics and negative entropy in de Sitter and anti-de Sitter Einstein-Gauss-Bonnet gravity,’’ Nucl. Phys. B 628 (2002), 295-330. [arXiv:hep-th/0112045 [hep-th]].
  * [32] T. Torii and H. Maeda, ‘‘Spacetime structure of static solutions in Gauss-Bonnet gravity: Neutral case,’’ Phys. Rev. D 71 (2005), 124002. [arXiv:hep-th/0504127 [hep-th]].
  * [33] R. A. Konoplya and A. Zhidenko, ‘‘Long life of Gauss-Bonnet corrected black holes,’’ Phys. Rev. D 82 (2010), 084003. [arXiv:1004.3772 [hep-th]].
  * [34] W. Xu, H. Xu and L. Zhao, ‘‘Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criticality,’’ Eur. Phys. J. C 74, 2970 (2014). [arXiv:1311.3053 [gr-qc]].
  * [35] H. Xu, W. Xu and L. Zhao, ‘‘Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions,’’ Eur. Phys. J. C 74, no. 9, 3074 (2014). [arXiv:1405.4143 [gr-qc]].
  * [36] H. Xu and Z. M. Xu, ‘‘Maxwell’s equal area law for Lovelock thermodynamics,’’ Int. J. Mod. Phys. D 26, no. 04, 1750037 (2016). [arXiv:1510.06557 [gr-qc]].
  * [37] Y. Sun, H. Xu and L. Zhao, ‘‘Thermodynamics and holographic entanglement entropy for spherical black holes in 5D Gauss-Bonnet gravity,’’ JHEP 1609, 060 (2016). [arXiv:1606.06531 [gr-qc]].
  * [38] R. Araneda, R. Aros, O. Miskovic and R. Olea, ‘‘Magnetic Mass in 4D AdS Gravity,’’ Phys. Rev. D 93 (2016) no.8, 084022. [arXiv:1602.07975 [hep-th]].
  * [39] R. A. Konoplya and A. Zhidenko, ‘‘Eikonal instability of Gauss-Bonnet–(anti-)–de Sitter black holes,’’ Phys. Rev. D 95 (2017) no.10, 104005. [arXiv:1701.01652 [hep-th]].
  * [40] R. A. Konoplya and A. Zhidenko, ‘‘Quasinormal modes of Gauss-Bonnet-AdS black holes: towards holographic description of finite coupling,’’ JHEP 09 (2017), 139. [arXiv:1705.07732 [hep-th]].
  * [41] D. Glavan and C. Lin, ‘‘Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime,’’  Phys. Rev. Lett. 124 (2020) no.8, 081301. [arXiv:1905.03601 [gr-qc]].
  * [42] R. G. Cai, L. M. Cao and N. Ohta, ‘‘Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black Hole Entropy,’’ JHEP 04 (2010), 082. [arXiv:0911.4379 [hep-th]].
  * [43] R. G. Cai, ‘‘Thermodynamics of Conformal Anomaly Corrected Black Holes in AdS Space,’’  Phys. Lett. B 733 (2014), 183-189. [arXiv:1405.1246 [hep-th]].
  * [44] G. Cognola, R. Myrzakulov, L. Sebastiani and S. Zerbini, ‘‘Einstein gravity with Gauss-Bonnet entropic corrections,’’ Phys. Rev. D 88 (2013) no.2, 024006 [arXiv:1304.1878 [gr-qc]].
  * [45] M. Guo and P. C. Li, ‘‘Innermost stable circular orbit and shadow of the $4D$ Einstein–Gauss–Bonnet black hole,’’ Eur. Phys. J. C 80 (2020) no.6, 588 [arXiv:2003.02523 [gr-qc]].
  * [46] P. G. S. Fernandes, ‘‘Charged Black Holes in AdS Spaces in $4D$ Einstein Gauss-Bonnet Gravity,’’ Phys. Lett. B 805 (2020), 135468. [arXiv:2003.05491 [gr-qc]].
  * [47] G. Anastasiou, O. Miskovic, R. Olea and I. Papadimitriou, ‘‘Counterterms, Kounterterms, and the variational problem in AdS gravity,’’ JHEP 08 (2020), 061 [arXiv:2003.06425 [hep-th]].
  * [48] R. Kumar and S. G. Ghosh, ‘‘Rotating black holes in $4D$ Einstein-Gauss-Bonnet gravity and its shadow,’’  JCAP 07 (2020) no.07, 053. [arXiv:2003.08927 [gr-qc]].
  * [49] C. Y. Zhang, P. C. Li and M. Guo, ‘‘Greybody factor and power spectra of the Hawking radiation in the $4D$ Einstein–Gauss–Bonnet de-Sitter gravity,’’ Eur. Phys. J. C 80 (2020) no.9, 874. [arXiv:2003.13068 [hep-th]].
  * [50] R. A. Konoplya and A. F. Zinhailo, ‘‘Grey-body factors and Hawking radiation of black holes in $4D$ Einstein-Gauss-Bonnet gravity,’’ Phys. Lett. B 810 (2020), 135793. [arXiv:2004.02248 [gr-qc]].
  * [51] C. Y. Zhang, S. J. Zhang, P. C. Li and M. Guo, ‘‘Superradiance and stability of the regularized 4D charged Einstein-Gauss-Bonnet black hole,’’ JHEP 08 (2020), 105. [arXiv:2004.03141 [gr-qc]].
  * [52] S. Mahapatra, ‘‘A note on the total action of 4D Gauss–Bonnet theory,’’ Eur. Phys. J. C 80 (2020) no.10, 992. [arXiv:2004.09214 [gr-qc]].
  * [53] J. Arrechea, A. Delhom and A. Jiménez-Cano, ‘‘Inconsistencies in four-dimensional Einstein-Gauss-Bonnet gravity,’’ Chin. Phys. C 45 (2021) no.1, 013107. [arXiv:2004.12998 [gr-qc]].
  * [54] K. Yang, B. M. Gu, S. W. Wei and Y. X. Liu, ‘‘Born–Infeld black holes in 4D Einstein–Gauss–Bonnet gravity,’’  Eur. Phys. J. C 80 (2020) no.7, 662. [arXiv:2004.14468 [gr-qc]].
  * [55] A. Kumar and R. Kumar, ‘‘Bardeen black holes in the novel $4D$ Einstein-Gauss-Bonnet gravity,’’ [arXiv:2003.13104 [gr-qc]].
  * [56] M. Gürses, T. Ç. Şişman and B. Tekin, ‘‘Is there a novel Einstein–Gauss–Bonnet theory in four dimensions?,’’  Eur. Phys. J. C 80 (2020) no.7, 647. [arXiv:2004.03390 [gr-qc]].
  * [57] B. Eslam Panah, K. Jafarzade and S. H. Hendi, ‘‘Charged 4D Einstein-Gauss-Bonnet-AdS black holes: Shadow, energy emission, deflection angle and heat engine,’’  Nucl. Phys. B 961 (2020), 115269. [arXiv:2004.04058 [hep-th]].
  * [58] J. Arrechea, A. Delhom and A. Jiménez-Cano, ‘‘Comment on “Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime”,’’ Phys. Rev. Lett. 125 (2020) no.14, 149002. [arXiv:2009.10715 [gr-qc]].
  * [59] M. Gurses, T. Ç. Şişman and B. Tekin, ‘‘Comment on ’’Einstein-Gauss-Bonnet Gravity in 4-Dimensional Space-Time’’,’’  Phys. Rev. Lett. 125 (2020) no.14, 149001. [arXiv:2009.13508 [gr-qc]].
  * [60] F. W. Shu, ‘‘Vacua in novel 4D Einstein-Gauss-Bonnet Gravity: pathology and instability?,’’ Phys. Lett. B 811 (2020), 135907. [arXiv:2004.09339 [gr-qc]].
  * [61] S. W. Hawking and D. N. Page, ‘‘Thermodynamics of Black Holes in Anti-de Sitter Space’’, Commun. Math. Phys. 87 (1983) 577.
  * [62] Y. C. Ong and Y. Yao, ‘‘Charged Particle Production Rate from Cosmic Censorship in Dilaton Black Hole Spacetimes,’’ JHEP 10 (2019), 129. [arXiv:1907.07490 [gr-qc]].
  * [63] D. Birmingham, ‘‘Topological black holes in Anti-de Sitter space’’, Class. Quant. Grav. 16 (1999), 1197-1205 . [arXiv:hep-th/9808032].
  * [64] P. Landsberg, A. De. Vos, ‘‘The Stefan-Boltzmann Constant in n-Dimensional Space’’, J. Phys. A: Math. Gen. 22 (1989) 1073.
  * [65] T. R. Cardoso and A. S. de Castro, ‘‘The Blackbody Radiation in D-Dimensional Universes’’, Rev. Bras. Ens. Fis. 27 (2005) 559 [arXiv:quant-ph/0510002].
  * [66] P. Chen, Y. C. Ong and D. h. Yeom, ‘‘Black Hole Remnants and the Information Loss Paradox,’’ Phys. Rept. 603 (2015), 1-45 [arXiv:1412.8366 [gr-qc]].
  * [67] R. Emparan and C. P. Herzog, ‘‘The Large D Limit of Einstein’s Equations’’, Rev. Mod. Phys. 92 (2020) no.4, 045005 [arXiv:2003.11394 [hep-th]].
  * [68] F. Holdt-Sørensen, D. A. McGady and N. Wintergerst, ‘‘Black Hole Evaporation and Semiclassicality At Large $D$’’, Phys. Rev. D 102 (2020) no.2, 026016 [arXiv:1908.08083 [hep-th]].
  * [69] T. Torii and H. Maeda, ‘‘Spacetime structure of static solutions in Gauss-Bonnet gravity: Charged case,’’ Phys. Rev. D 72 (2005), 064007 [arXiv:hep-th/0504141 [hep-th]].
  * [70] G. Penington, ‘‘Entanglement Wedge Reconstruction and the Information Paradox,’’ JHEP 09 (2020), 002 [arXiv:1905.08255 [hep-th]].
  * [71] A. Almheiri, R. Mahajan and J. E. Santos, ‘‘Entanglement islands in higher dimensions,’’ SciPost Phys. 9 (2020) no.1, 001 [arXiv:1911.09666 [hep-th]].
  * [72] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, ‘‘The entropy of Hawking radiation,’’ [arXiv:2006.06872 [hep-th]].


# Superconductivity in cuprates governed by topological constraints

Yves Noat  Institut des Nanosciences de Paris, CNRS, UMR 7588
Sorbonne Université, Faculté des Sciences et Ingénierie, 4 place Jussieu,
75005 Paris, France  Alain Mauger  Institut de Minéralogie, de Physique des
Matériaux et de Cosmochimie, CNRS, UMR 7590,Sorbonne Université, Faculté des
Sciences et Ingénierie, 4 place Jussieu, 75005 Paris, France  William Sacks
Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, CNRS,
UMR 7590,Sorbonne Université, Faculté des Sciences et Ingénierie, 4 place
Jussieu, 75005 Paris, France

###### Abstract

The remarkable universality of the cuprate $T_{c}$ dome suggests a very
fundamental unifying principle. Moreover, the superconducting gap is known to
persist above $T_{c}$ in the pseudogap phase of all cuprates. So, contrary to
BCS, the gap cannot be the order parameter of the transition.

In this work, we show that both the $T_{c}$-dome and the pseudogap line
$T^{*}(p)$ arise from a unique and identifiable principle: the interaction of
localized ‘pairons’ on an antiferromagnetic square lattice. The topological
constraints on such preformed pairons give rise to both the $T_{c}$ dome and
the pairing energy simultaneously. It also provides a natural explanation for
the critical doping points of the phase diagram.

The model matches perfectly both the $T^{*}$ and $T_{c}$ experimental lines,
with only one adjustable parameter.

###### pacs:

74.72.h,74.20.Mn,74.20.Fg

##  .1 Introduction

Despite more than thirty years of intense research and many advances, a
general understanding of the physics of cuprates is still lacking. Indeed,
most of the key questions remain to be answered or clarified:

  1. i)

What is the pairing mechanism?

  2. ii)

What is the nature of the SC condensation?

  3. iii)

What is the nature of the pseudogap phase and its connection with the SC
state?

  4. iv)

What is the physical origin of the critical doping points?

Figure 1: (Color online) Essential phase diagram of cuprates (see [2] for
details) with the three characteristic temperature lines: the critical
temperature $T_{c}(p)$, the pseudogap line $T^{*}(p)$ and the characteristic
temperature of magnetic correlations, $T_{max}(p)$.

Superconductivity emerges in cuprates from doping an antiferromagnetic (AF)
Mott insulator, leading to a complex phase diagram as shown in Fig. 1. The
latter displays three important lines : the superconducting dome $T_{c}(p)$,
the pseudogap line $T^{*}(p)$ and the characteristic temperature of magnetic
correlations $T_{max}(p)$.

Despite the very large variety of compounds belonging to the cuprate family,
there is a striking universality in the phase diagram. In addition, as already
noted by Tahir et al. [1], the doping values characterizing the $T_{c}$-dome
($p_{min}$, the onset doping for superconductivity, $p_{opt}$ the top of the
dome or optimal doping and $p_{max}$ the end of the dome) seem to be universal
values, which are practically independent of the material.

All these experimental facts strongly suggest that superconductivity in
cuprates is essentially governed by topological constraints imposed on the
system of holes by the 2d antiferromagnetic square lattice.

##  .2 Magnetic properties

Contrary to conventional superconductors which are mostly ‘good’ metals,
parent compounds of SC cuprates are antiferromagnetic Mott insulators,
characterized by the temperature of long-range magnetic ordering
$T_{\textrm{\tiny N\'{e}el}}(p)$. Electron or hole doping strongly modifies
the electronic and magnetic properties. First, the Néel temperature decreases
rapidly with doping and finally vanishes for a small value $p\approx$0.015
[3]. Then, above $p_{min}=$0.05, the system becomes metallic and exhibits
superconducting properties below the critical temperature $T_{c}$.

Measurements of the magnetic susceptibility $\chi(T)$ [4, 5, 6, 7, 8] have
shown that the characteristic temperature of magnetic correlations,
$T_{max}$(defined as the peak in the magnetic susceptibility as a function of
temperature) decreases with doping. In a previous work [2], we have shown that
a similar behavior is observed in the four different compounds (La2-xSrxCuO4,
Bi2Sr2Ca1-xYxCu2O8, Bi2Sr2CaCu2O8+y, YBa2Cu3O6+y): $T_{max}(p)$ decreases
linearly with $p$ over a wide range and then saturates in the overdoped
regime. In addition, the extrapolation of the linear behavior to zero
temperature gives a critical doping $p_{c}\sim$0.2.

Figure 2: (Color online) Phase diagram in the pairon-simplon viewpoint. In the
underdoped regime, a simple picture emerges: simplons (right panel, a) exist
below the line $T_{max}(p)$ while incoherent pairons (right panel, b) exist
below the pseudogap line $T^{*}(p)$. Finally, pairons condense below
$T_{c}(p)$ in the coherent SC state. Three key doping points are illustrated:
$p_{min}$, the onset doping of superconductivity, $p_{max}$, the end of the
$T_{c}$-dome, and $p_{c}\sim$0.2 the extrapolated value of the line
$T_{max}(p)$ at $T=0$.

The linear behavior of $T_{max}(p)$ with hole density can be qualitatively
understood by considering that each hole ‘suppresses’ the four nearest
neighbors spins. We call this dressed hole a ‘simplon’, which is an effective
particle defined as a hole with 4 associated neighboring sites, thus giving
4+1 ‘spinless’ sites (see Fig. 2, right panel). A simple density argument
leads to the linear law

$k_{B}T_{max}\simeq J(1-5p)$ (1)

where $J$ is the magnetic exchange energy. The critical doping value
$p_{c}=$0.2, where $T_{max}$ vanishes, corresponds to the compact simplon
lattice (Fig. 3), with a superlattice constant $d=\sqrt{5}$. Interestingly,
this doping value is very close to the one which has been identified either as
the end of the pseudogap [9] or a quantum critical point (QCP) [10].

Figure 3: (Color online) The compact simplon lattice (yellow square) with
superlattice constant $d=\sqrt{5}$, corresponds to the ‘critical’ doping value
$p_{c}=$0.2. The two sublattices of equivalent simplons (A or B, green square)
have a lattice constant $d_{0}=\sqrt{10}$, giving the doping value $p=$0.1.
The doping $p_{min}=$0.05 corresponds to a simplon on the corner of the red
(non-centered) square. We use $a_{0}$ units throughout.

Two types of simplons, A and B, can be distinguished depending on the spin
state of the four electrons surrounding a hole. The unit cell of the square
sublattice of equivalent simplons A (or B) has a side $d=\sqrt{10}$,
corresponding to a density $p=$0.1 (see Fig. 3). The percolation point of
equivalent simplons on such a sublattice is the previous density divided by
two, i.e. $p=$0.05. This is very close to the experimental value of $p_{min}$,
the onset doping for superconductivity. Metallicity could arises
preferentially due to the coupling of equivalent simplons (A or B), which can
interact provided their distance is smaller than $d_{0}\approx\sqrt{20}$.

##  .3 Pairing in cuprates

In conventional superconductors, a bound state of electrons can form in the
electron sea as a result of phonon exchange. Cooper pairs [11] are delocalized
objects which are well described in $k$-space as pairs of opposite wave
vectors and spins, $\left|\vec{k}\uparrow-\vec{k}\downarrow\right\rangle$.

We proposed in Ref. [12] that the pairing mechanism in cuprate belongs to a
completely different class. The magnetic ordering of the Néel state is
destroyed by hole doping but survives on the local scale. As shown by
Birgeneau et al. by neutron measurements in Ref. [13], the antiferromagnetic
coherence length varies roughly as the average distance between hole
$\xi_{AF}\sim a/\sqrt{p}$. In our model [12, 14], below the characteristic
temperature $T^{*}$, two adjacent holes tend to form a bound state due to this
local AF environment, with a binding energy on the order of $J$, as confirmed
by early numerical calculations with the Hubbard or t-J hamiltonian [15, 16,
17, 18, 19]. Thus, a new kind of pair exists in cuprates, pairs of holes or
‘pairons’ in their local AF environment, in real space on a typical length
$\xi_{AF}$. The existence of pairons is supported by the detailed angular
dependence of the gap function [14] measured by Angular Resolved Photoemission
Spectroscopy (ARPES) [20, 21].

Electrons within the AF coherence length contribute to the pairing energy of
the two holes. As the doping increases, the AF coherence length decreases as
$\sim 1/\sqrt{p}.$ A pairon can exist provided two adjacent holes are
surrounded by at least a ring of six electrons. The pairons thus occupy a
minimum of 8 sites (2 holes plus six electrons) on the square lattice in the
local AF environnement.

Figure 4: (Color online) Upper panel: Schematic of the pairon lattice for
three different doping values corresponding to a) underdoped regime, b)
optimal doping, c) maximum doping (compact pairon lattice).

This topological constraint imposes a maximum doping value corresponding to
the compact pairon lattice (see Fig. 4), which is obtained for p=1/4. This
value is remarkably close to the observed $p_{max}=$0.27 at the dome limit.
Actually, experimental facts indicate that pairons can still exist for a
slightly larger doping value 0.25$+\delta$, up to $p_{max}=$0.27. The reason
for this small excess is unknown and requires further thought, but the
experimental value corresponds to an additional 1/6 hole per pairon ring
($\delta=x/8=1/48\approx$0.02). Nevertheless, the compact pairon lattice
provides a satisfactory topological interpretation.

As previously mentioned, the distance between pairons is of the order of the
spin correlation length $\xi_{AF}$. The hypothesis that only sites within the
correlation length contribute to the pairon binding energy implies that the
binding energy, or equivalently the temperature of pairon formation $T^{*}$,
varies linearly as a function of doping [12]:

$T^{*}\propto(1-4p).$ (2)

This linear law is accurately confirmed by ARPES [22, 23] and tunneling
measurements [24, 25].

Both the $T_{max}(p)$ and $T^{*}(p)$ lines are thus given by similar
topological arguments based on two fundamental quantum objects, simplons and
pairons.

##  .4 Superconducting condensation

In conventional superconductors, the gap $\Delta(T)$ is the order parameter
[26]. At finite temperature, the SC state is gradually destroyed by means of
quasiparticle fermionic excitations: the pair-breaking mechanism.
Consequently, the gap $\Delta(T)$ decreases with increasing temperature and
finally vanishes at the critical temperature $T_{c}$. These considerations are
captured by the BCS relation [26]: $1.7k_{B}T_{c}=\Delta(0)$.

The situation must be different in cuprates since, as well demonstrated
unequivocally by tunneling spectroscopy [27, 28] and ARPES [29, 23]
measurements, the gap clearly does not vanish at $T_{c}$ and a pseudogap
remains up to the higher temperature $T^{*}$. The conventional BCS relation is
therefore no longer valid. What is then the order parameter in cuprates?

In several articles [30, 12], we have proposed that, unlike conventional SC,
condensation in cuprates arises because of pairon-pairon interactions.
Moreover, the fundamental excitations of the condensate are pairon excitations
governed by Bose statistics [31, 32]. A simple picture emerges (see Fig. 2):
$T_{max}$ corresponds to the characteristic temperature of magnetic
correlations below which simplons are formed. Below $T^{*}$ pairons are
formed, corresponding to the pseudogap state where they remain incoherent.
Finally, below $T_{c}$, pairons condense in the coherent SC state.

Therefore, in our model, the total energy $E_{SC}$ of the superconducting
state (per pair) is unconventional and reads [32]:

$E_{SC}=-\Delta_{p}-\beta_{c}$ (3)

where $\Delta_{p}$ is the zero-temperature binding energy and $\beta_{c}$ is
the condensation energy responsible for long range order. The latter can be
precisely extracted from the experimental quasiparticle spectra [33, 30].
Contrary to BCS, since $\Delta_{p}$ is constant across $T_{c}$, it is
$\beta_{c}$ that determines the critical temperature and not the gap [30],
with the result:

$\beta_{c}\simeq 2.2k_{B}T_{c}$ (4)

We now show that this condensation energy $\beta_{c}$, i.e. the energy
difference between the non SC pseudogap state $E_{PG}=-\Delta_{p}$ and the SC
state $E_{SC}$, results from quantifying the amount of pairon disorder. In
this view, the characteristic disorder in the PG state is uniquely determined
by the topological constraints of the 2d square lattice.

Figure 5: (Color online) Statistics of the number of pairons $N_{i}$ in a
square of size $d_{0}=5$ containing $N_{0}=$25 possible sites. Left panel :
Mean value $\left\langle N_{i}\right\rangle$ and variance
$\sigma^{2}=\left\langle N_{i}^{2}\right\rangle-\left\langle
N_{i}\right\rangle^{2}$. plotted as a function of nominal pairon density
$p^{\prime}$. Middle panel : Histogram $H(N_{i})$ of the number of pairons
$N_{i}$ for $p^{\prime}=$0.22, chosen as an example. Right panel: Pairons
randomly distributed on the pairon lattice for $p^{\prime}=$0.22.

To proceed, akin to a lattice gas approach, let us consider pairons on an
equivalent square lattice of density $p^{\prime}$: $p^{\prime}=$1 corresponds
to the compact pairon lattice ($p=$0.27 on the Cu0 square lattice) and
$p^{\prime}=$0 to the onset doping at which the average distance between
pairons is small enough ($d<d_{0}$) so that they interact ($p=$.05 on the Cu0
square lattice). Pairons are then randomly distributed on this lattice (Fig.
5), under the hypothesis where all sites are equivalent. We calculate the
number of pairons, $N_{i}$, inside a square of side $d_{0}$ (with
$N_{0}={d_{0}}^{2}$ sites) , where $d_{0}$ is the maximum interacting distance
between pairons. From the distribution of $\left\langle N_{i}\right\rangle$,
we calculate the statistical averages, the mean value $\left\langle
N_{i}\right\rangle$ and the variance $\sigma^{2}=\left\langle
N_{i}^{2}\right\rangle-\left\langle N_{i}\right\rangle^{2}$.

As expected for a problem depending only on the success $p^{\prime}$ or
failure $1-p^{\prime}$ to find a pairon on a given site, we obtain for $N_{i}$
a binomial distribution. As a result, $\left\langle N_{i}\right\rangle$
follows a straight line as a function of $p^{\prime}$, with some jitter, while
the variance $\sigma^{2}=N_{0}p^{\prime}(1-p^{\prime})$ displays a dome shape.
In fact, for a fixed $N_{0}$ and a random distribution, $\sigma^{2}$
characterizes the maximum disorder of the localized pairons on the square
lattice.

The condensation energy can be understood using the Following gendanken
experiment. When the interaction between pairons is switched off, we obtain
the incoherent PG state, the ‘pairon glass’, of energy $E_{PG}$, where there
is by definition no correlation between pairons. In this state, no long-range
SC order exists and the disorder is described by the above binomial
distribution.

When the interaction is turned back on (provided their typical distance $d$ is
smaller than $d_{0}$), all pairons are in the ordered SC ground state, with
energy $E_{PG}-\beta_{c}$. In this zero temperature transformation, the
virtual work $W$ needed to disorder the system is $\beta_{c}$:

$W=E_{PG}-E_{SC}=\beta_{c}$ (5)

Since the disorder is characterised by the variance of the distribution, one
should have $W\propto\sigma^{2}$. Thus, we obtain

$\beta_{c}\propto\sigma^{2}/N_{0}=p^{\prime}(1-p^{\prime}).$ (6)

In this view, the condensation is a new type of disorder to order transition,
independent of the other degrees of freedom. In spite of the complex
excitations, Bose excitations and quasiparticle fermionic excitations, and
magnetic degrees of freedom, the underlying topological constraints define the
fundamental mechanism.

##  .5 Pseudogap and superconductivity

We now focus on the interplay between the pseudogap and the superconducting
phase in cuprate. The pseudogap line is given by $T^{*}\propto(1-4p)$, which
translates to $\sim(1-p^{\prime})$ in the pairon sublattice, and that the
condensation energy is $\beta_{c}\sim p^{\prime}(1-p^{\prime})$. Furthermore,
the statistical calculation described above leads the two relations:

$\displaystyle T^{*}=\lambda\,(1-p^{\prime})$ $\displaystyle
T_{c}=\lambda\,p^{\prime}(1-p^{\prime})$ (7)

where $\lambda$ has the dimension of temperature. The true doping value is
given by the linear tranformation $p=p_{min}+(p_{max}-p_{min})\times
p^{\prime}$.

Figure 6: (Color online) (a) Antinodal gap (in units of $\Delta/2.2k_{B}$) (b)
pseudogap temperature $T^{*}$, both measured by ARPES [23, 34] and (c)
experimental superconducting temperature $T_{c}$ (from Ref. [9]) , compared to
the binomial law (red lines) with only one adjustable parameter $\lambda$.
From the fit of the data we find the value $\lambda=$400K.

To proceed, we now compare in Fig. 6 the calculated $T^{*}$ and $T_{c}$ to the
experimental values, measured by ARPES [34] and resistivity [9] respectively.
With only one adjustable parameter $\lambda$, the agreement between theory and
experiments is remarkable. We find the value $\lambda=$400$K$, which is
comparable to $J/2$, or the Néel temperature when extrapolated to $p=0$.

The binomial law provides a simple explanation for the maximum $T_{c}$
obtainable, which occurs at half-filling, $p^{\prime}=1/2$. Indeed, at this
concentration the fluctuation amplitude, measured by $\sigma^{2}$, is a
maximum. At this optimum concentration, we see the topological constraint that
$T^{*}$ is exactly twice $T_{c}$.

These fundamental relations illustrate that the pseudogap and
superconductivity are completely linked and arise from the same physical
phenomenon, for the entire doping range. The geometry of the dome and the
tangent line are not arbitrary but determined by the statistics of pairons
randomly distributed on a square lattice. A direct consequence is that both
$T^{*}$ and $T_{c}$ are proportional to the same energy scale, which we
identify as the magnetic exchange energy $J$.

##  .6 Conclusion

In this article, we propose that key aspects of the phase diagram of cuprates
are governed by the topological properties of the doped antiferromagnetic
insulator on a square lattice. Bound pairs of holes are formed due to the
local antiferromagnetic environnement and condense in the superconducting
state in a disorder to order transition. The condensation is driven by a new
mechanism and is directly related to the amount of disorder in the non SC
pseudogap state. Thus the simple binomial pairon distribution explains the
$T_{c}$-dome as well as the pseudogap line. We show that the cuprate SC state,
a spatially correlated quantum state of pairons, is intimately connected to
the disordered pseudogap state – in our view they appear as indissociable
phenomena.

##  .7 Acknowledgments

The authors gratefully acknowledge many fruitful discussions with H. Eisaki
and S. Ishida (AIST, Tsukuba, Japan) and M. Nohara (University of Hiroshima,
Japan).

## References

  * [1] Jamil Tahir-Kheli and William A. Goddard III, Universal Properties of Cuprate Superconductors: Tc Phase Diagram, Room-Temperature Thermopower, Neutron Spin Resonance, and STM Incommensurability Explained in Terms of Chiral Plaquette Pairing, J. Phys. Chem. Lett. 1, 1290–1295 (2010).
  * [2] Y. Noat, A. Mauger, M. Nohara, H. Eisaki, W. Sacks, Cuprates phase diagram deduced from magnetic susceptibility: what is the ’true’ pseudogap line?, article submitted to Solid State communications (2022).
  * [3] B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y. Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh, R. W. Erwin, and G. Shirane, Magnetic excitations in pure, lightly doped, and weakly metallic La2CuO4, Phys. Rev. B 46, 14034 (1992).
  * [4] David C. Johnston, Magnetic Susceptibility Scaling in La2-xSrxCuO4-y, Phys. Rev. Lett. 62, 957 (1989).
  * [5] J. B. Torrance, A. Bezinge, A. I. Nazzal, T. C. Huang, S. S. P. Parkin, D. T. Keane, S. J. LaPlaca, P. M. Horn, and G. A. Held, Properties that change as superconductivity disappears at high-doping concentrations in La2-xSrxCuO4, Phys. Rev. B 40, 8872 (1989).
  * [6] R. Yoshizaki, N. Ishikawa, H. Sawada, E. Kita, A. Tasaki, Magnetic susceptibility of normal state and superconductivity of La2-xSrxCuO4, Physica C 166, 417 (1990).
  * [7] M.Oda, T.Nakano, Y.Kamada, M.Ido, Electronic states of doped holes and magnetic properties in La2?xMxCuO4 (M = Sr, Ba), Physica C 183, 234 (1991).
  * [8] T. Nakano, M. Oda, C. Manabe, N. Momono, Y. Miura, and M. Ido, Magnetic properties and electronic conduction of superconducting La2-xSrxCuO4, Phys. Rev. B 49, 16000 (1994).
  * [9] J. L. Tallon and J.W. Loram, The doping dependence of $T^{*}$ \- what is the real high-$T_{c}$ phase diagram?, Physica C 349, 53 (2001).
  * [10] C. Proust and L. Taillefer, Annual Review of Condensed Matter Physics 10, 409 (2019).
  * [11] Leon N. Cooper, Bound electron pairs in a degenerate Fermi gas, Physical Review 104, 1189-1190 (1956).
  * [12] W. Sacks, A. Mauger and Y. Noat, Cooper pairs without glue in high-$T_{c}$ superconductors: A universal phase diagram, Euro. Phys. Lett 119, 17001 (2017).
  * [13] R. J. Birgeneau, D. R. Gabbe, H. P. Jenssen, M. A. Kastner, P. J. Picone, T. R. Thurston, G. Shirane, Y. Endoh, M. Sato, K. Yamada, Y. Hidaka, M. Oda, Y. Enomoto, M. Suzuki, and T. Murakami, Antiferromagnetic spin correlations in insulating, metallic, and superconducting La2-xSrxCuO4, Phys. Rev. B 38, 6614 (1988).
  * [14] Y. Noat, A. Mauger and W. Sacks, Single origin of the nodal and antinodal gaps in cuprates, Euro. Phys. Lett 126, 67001 (2019).
  * [15] Efthimios Kaxiras and Efstratios Manousakis, Hole dynamics in the two-dimensional strong-coupling Hubbard Hamiltonian, Phys. Rev. B 38, 866(R) (1988).
  * [16] J. Bonča, P. Prelovšek, and I. Sega, Exact-diagonalization study of the effective model for holes in the planar antiferromagnet, Phys. Rev. B 39, 7074 (1989).
  * [17] J. A. Riera and A. P. Young, Binding of holes in one-band models of oxide superconductors, Phys. Rev. B 39, 9697(R) (1989).
  * [18] Y. Hasegawa and D. Poilblanc, Hole dynamics in the t-J model: An exact diagonalization study, Phys. Rev. B 40, 9035 (1989).
  * [19] Didier Poilblanc, José Riera and Elbio Dagotto, d-wave bound state of holes in an antiferromagnet, Phys. Rev. B 49, 12318 (1994).
  * [20] K. Terashima, H. Matsui, T. Sato, T. Takahashi, M. Kofu, and K. Hirota, Anomalous Momentum Dependence of the Superconducting Coherence Peak and Its Relation to the Pseudogap of La1.85Sr0.15CuO4, Phys. Rev. Lett. 99, 017003 (2007).
  * [21] H. Anzai, A. Ino, M. Arita, H. Namatame, M. Taniguchi, M. Ishikado, K. Fujita, S. Ishida and S. Uchida, Relation between the nodal and antinodal gap and critical temperature in superconducting Bi2212, Nature Communications 4, 1815 (2013).
  * [22] S. Hüfner, M. A. Hossain, A Damascelli, and G. A. Sawatzky,Two gaps make a high-temperature superconductor?, Rep. Prog. Phys., 71, 062501 (2008).
  * [23] Makoto Hashimoto, Inna M. Vishik, Rui-Hua He, Thomas P. Devereaux and Zhi-Xun Shen, Energy gaps in high-transition-temperature cuprate superconductors, Nature Physics 10, 483 (2014).
  * [24] Tohru Nakano, Naoki Momono, Migaku Oda, and Masayuki Ido, Correlation between the Doping Dependences of Superconducting Gap Magnitude $2\Delta_{0}$ and Pseudogap Temperature $T^{*}$ in High–$T_{c}$ Cuprates, J. Phys. Soc. Jpn. 67, 2622 (1998).
  * [25] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod and C. Renner, Scanning tunneling spectroscopy of the cuprates, Rev. Mod. Phys. 79, 353 (2007).
  * [26] J. Bardeen, L. Cooper, J. Schrieffer, Theory of Superconductivity, Phys. Rev. 108 1175 (1957).
  * [27] Ch. Renner,B. Revaz, J.-Y. Genoud, K. Kadowaki,and Ø. Fischer, Pseudogap precursor of the superconducting gap in under- and overdoped Bi2Sr2CaCu2O8+δ, Phys. Rev. Lett., 80 149 (1998).
  * [28] R. Sekine, S. J. Denholme, A. Tsukada, S. Kawashima, M. Minematsu,T. Inose, S. Mikusu, K. Tokiwa, T. Watanabe, and N. Miyakawa, Characteristic features of the mode energy estimated from tunneling conductance on TlBa2Ca2Cu3O8.5+δ, J. Phys. Soc. Jpn. 85, 024702 (2016).
  * [29] H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria, M. R. Norman, T. Mochiku, K. Kadowaki and J. Giapintzakis, Spectroscopic evidence for a pseudogap in the normal state of underdoped high–$T_{c}$ superconductors, Nature 382, 51 (1996).
  * [30] W. Sacks, A. Mauger, Y. Noat, Pair – pair interactions as a mechanism for high-Tc superconductivity, Superconduct. Sci. Technol., 28 105014 (2015).
  * [31] W. Sacks, A. Mauger and Y. Noat, Origin of the Fermi arcs in cuprates: a dual role of quasiparticle and pair excitations, Journal of Physics: Condensed Matter, 30, 475703 (2018).
  * [32] Y. Noat, A. Mauger, M. Nohara, H. Eisaki, W. Sacks , How ‘pairons’ are revealed in the electronic specific heat of cuprates, Solid State Communications 323, 114109 (2021).
  * [33] W. Sacks, T. Cren, D. Roditchev, and B. Douçot,Quasiparticle spectrum of the cuprate Bi2Sr2CaCu2O8+δ: Possible connection to the phase diagram, Phys. Rev. B 74, 174517 (2006).
  * [34] I. M. Vishik, M. Hashimoto, R.-H. He, W.-S. Lee, F. Schmitt, D. Lu, R. G. Moore, C. Zhang, W. Meevasana, T. Sasagawa, S. Uchida, Kazuhiro Fujita, S. Ishida, M. Ishikado, Y. Yoshida, H. Eisaki, Z. Hussain, T. P. Devereaux, and Z.-X. Shen, Phase competition in trisected superconducting dome, PNAS 109, 18332 (2012).