Page Curve and Phase Transition in deformed Jackiw-Teitelboim Gravity

Cheng-Yuan Lu1, Ming-Hui Yu1, Xian-Hui Ge1∗, Li-Jun Tian1††footnotetext: *
Corresponding author<EMAIL_ADDRESS>

1Department of Physics, Shanghai University, Shanghai 200444, China

###### Abstract

We consider the entanglement island in a deformed Jackiw-Teitelboim black hole
in the presence of the phase transition. This black hole has the van der
Waals-Maxwell-like phase structure as it is coupled with a Maxwell field. We
study the behavior of the Page curve of this black hole by using the island
paradigm. In the fixed charge ensemble, we discuss different situations with
different charges that influence the system’s phase structure. There is only a
Hawking-Page phase transition in the absence of charges, which leads to an
unstable small black hole. Hence, the related Page curve does not exist.
However, a van der Waals-Maxwell-like phase transition occurs in the presence
of charges. This yields three black hole solutions. The Page curve of the
middle size black hole does not exist. For the extremal black hole, the Page
time approaches zero in the phase transition situation but becomes divergent
without the phase transition. In a word, we study the Page curve and the
island paradigm for different black hole phases and in different phase
transition situations.

###### Contents

  1. 1 Introduction
  2. 2 Set up
  3. 3 Entanglement Entropy
    1. 3.1 Cutoff surface far away horizon
    2. 3.2 Cutoff surface near horizon
  4. 4 Page curve and Page time
  5. 5 Phase transition of the black hole
    1. 5.1 $Q=0$
    2. 5.2 $Q\neq 0$
  6. 6 Discussion and Conclusion

##  1 Introduction

It is widely believed that studying the information loss paradox is
significant for revealing quantum gravity. In 1975, Stephen Hawking discovered
that the black hole would emit black-body radiation which does not carry any
information about the interior of the black hole [1, 2]. If a black hole is
formed by the pure state, then the whole process of black hole evaporation
corresponds to the evolution from the pure state to the mixed state, which
breaks the unitary principle of quantum mechanics. If quantum mechanics is
respected, the fine-grained entropy, also called entanglement entropy or von-
Neuman entropy, of Hawking radiation should drop to zero at the end of the
whole process. However, Don Page pointed out that if black hole evaporation is
a unitary process, then the entanglement entropy of the radiation should
linearly increase at the early time, and decrease after a special time – Page
time, this curve is called the Page curve [3, 4]. Recently, there is a great
breakthrough in this issue. By using the island formula [5], the Page curve
emerges naturally. It leads to a decreasing fine-grained entropy for the
Hawking radiation due to the emergence of the island in the black hole
interior. The fine-grained entropy of Hawking radiation can be expressed as
[6, 7]

$S_{\text{rad}}=\text{Min}_{I}\left\\{\text{Ext}_{I}\left[\frac{\text{Area}(\partial
I)}{4G_{\mathrm{N}}}+S_{\mathrm{vN}}\left(R\cup I\right)\right]\right\\},$
(1.1)

where $I$ represents the island, $G_{N}$ is the Newton constant, $R$ is the
radiation region and $\partial I$ is the boundary of the island. Crucially,
the calculations include the contribution of islands inside the black hole to
the radiation entropy. In order to obtain the right answer, we should first
extremize the generalized entropy and then pick up the minimum value to match
this formula. Before the Page time, there are no islands, so the generalized
entropy without the island is the smaller one. However, the generalized
entropy with the island has more weight in the process of minimization after
the Page time. Therefore, the entanglement wedge not only includes the
outgoing Hawking radiation, but also the black hole interior purifying
partner.

The island formula was first proposed in a solvable two dimensional
evaporating black hole model which is constructed by coupling a bath to a
nearly-$\rm{AdS_{2}}$ black hole [8]. Different from the evaporating black
hole, the island of the eternal black hole is well established and outside the
event horizon [9]. Although the island formula is derived from Jackiw-
Teitelboim(JT) black holes, its application is beyond the context of Anti-de
Sitter($\rm{AdS}$) spacetime. It can be applied to other spacetime
backgrounds, such as the other $\text{AdS}_{2}$ black hole [10, 11, 12, 13,
14, 15, 16, 17], the two-dimensional(2D) dilaton gravity [18, 19, 20, 21, 22,
23, 24], the higher-dimensional spacetime [25, 26, 28, 29, 30, 31, 32, 33, 27,
34], the cosmology [35, 36, 37, 38, 39, 40, 41, 42], the boundary conformal
field theory(BCFT) [43, 44, 45, 46, 47, 48, 49]. Interestingly, there are also
some related works for the Sachdev-Ye-Kitaev(SYK) model [50, 51, 52, 53, 54,
55].

Up to now, most studies have not yet considered the influence of the black
hole phase transition on the Page curve. Considering the possibility that the
phase transition may change the structure of the Page curve, we would like to
study the relationship among the island formula, the Page curve, and the phase
transition. Based on this motivation, this paper studies a special 2D black
hole solution that yields van der Waals-Maxwell-like phase structure in
deformed JT gravity [62, 63, 64]. Following [9], we consider a two-side
eternal black hole coupled to flat thermal baths. Then we use these strategies
to obtain the Page curve and discuss the effect of the phase transition on the
Page curve.

This paper is organized as follows. In section 2, we discuss the black hole
solution which is derived from JT-Maxwell gravity and introduce our toy model
– the two-sided deformed JT black hole coupled thermal baths. In section 3, we
calculate the evolution of entanglement entropy without island and display the
paradox. Then, we consider an island that would avoid the divergence of the
entanglement entropy at late times. In section 4, based on the previous
result, we plot the Page curve and derive the scrambling time. In section 5,
we considered these results in combination with the phase transition. We
discuss the change of the Page time with different charges in the canonical
ensemble. We consider both the Hawking-Page phase transition without charges
and the van der Waals-Maxwell-like phase transition with charges. Finally, we
give a summary and some remarks in the last section. Hereafter,we use the unit
$\hbar=k_{B}=c=1$.

##  2 Set up

In this section, we briefly review the deformed JT gravity with a Maxwell
field [58, 59, 60, 61, 62]. We can write down the action for the deformed JT
gravity with metric $g$ and dilaton coupled to conformal matter, the action
can be written as follows [56, 57],

$\displaystyle S[g,\phi,\Phi]$
$\displaystyle=S_{\mathrm{top}}[g]+S_{\mathrm{JT}}[g,\phi]+S_{\rm{m}}[g,\Phi]$
(2.1) $\displaystyle=\frac{\phi_{0}}{16\pi
G_{\mathrm{N}}}\left[\int_{\mathcal{M}}\sqrt{-g}R+2\int_{\partial\mathcal{M}}\sqrt{-\gamma}K\right]$
$\displaystyle+\frac{1}{16\pi
G_{\mathrm{N}}}\left[\int_{\mathcal{M}}\sqrt{-g}\phi(R+2)+2\int_{\partial\mathcal{M}}\sqrt{-\gamma}\phi_{b}(K-1)\right]$
$\displaystyle+S_{\rm{m}}[g,\Phi].$

The first term $S_{\rm{top}}$ is a purely topological term and only
contributes a constant $\phi_{0}\chi$, where $\chi$ is the Euler
characteristic of the corresponding manifold $\mathcal{M}$. The second term
$S_{\rm{JT}}$ is the consequence of dimensional reduction. The dilaton $\phi$,
the Gibbons-Hawking-York boundary term, a counter-term with boundary metric
$\gamma$ and curvature $K$, and boundary dilaton value $\phi_{\rm{b}}$ are
included in $S_{\rm{JT}}$. The last term $S_{\rm{m}}$ is simply a 2D CFT
action for a matter field, which does not couple to the dilaton directly.

Hereafter, we consider a deformed JT gravity plus a Maxwell field. The action
for this theory reads as [62]

$\displaystyle S_{\mathrm{JT}}^{\text{charg }}$
$\displaystyle=\frac{\phi_{0}}{2}\left(\int_{\mathcal{M}}d^{2}x\sqrt{-g}R+2\int_{\partial
M}\sqrt{-\gamma}Kd\tau\right)$ (2.2)
$\displaystyle+\frac{1}{2}\int_{\mathcal{M}}d^{2}x\sqrt{-g}\left(\phi
R+\frac{V(\phi)}{l^{2}}-\frac{1}{2}Z(\phi)\mathcal{F}^{2}\right)$
$\displaystyle+\frac{1}{2}\int_{\partial\mathcal{M}}d\tau\sqrt{-\gamma}K_{\mu}Z(\phi)\mathcal{F}^{\mu\nu}\mathcal{A}_{\nu}+\int_{\partial\mathcal{M}}\sqrt{-\gamma}\phi_{b}Kd\tau$
$\displaystyle+\int_{\partial\mathcal{M}}\sqrt{-\gamma}\mathcal{L}_{\mathrm{ct}}d\tau,$

where $l$ is the Ads radius, $\mathcal{F}$ is the Maxwell field and
$\mathcal{L}_{\mathrm{ct}}$ is the counterterms. The first bracket is the
topological term for JT gravity. Through the variation of this action, we can
obtain the black hole solution of the above theory in the Schwarzschild
coordinates,

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)},$ (2.3a) $\displaystyle
V(\phi)=2\phi+\frac{a_{0}}{\phi^{\eta}},\quad\phi(r)=r,$ (2.3b) $\displaystyle
f(r)=\frac{r^{2}-r_{\rm{H}}^{2}}{l^{2}}+\frac{a_{0}}{(1-\eta)l^{2}}\left(r^{1-\eta}-r_{\rm{H}}^{1-\eta}\right)-\frac{Q^{2}}{(1-\zeta)l^{2}}\left(r^{1-\zeta}-r_{\rm{H}}^{1-\zeta}\right),$
(2.3c) $\displaystyle
Z(\phi)=r^{\zeta},\quad\mathcal{A}_{t}(r)=\mu+\frac{Q}{1-\zeta}r^{1-\zeta},$
(2.3d)

where $l$ is the AdS radius, $r_{\rm{H}}$ is the event horizon of the black
hole, $Q$ represents the charge density, and the parameters satisfy
$a_{0}>0,\eta>0,\zeta>0$. The dilaton potential $V\left(\phi\right)$ satisfies
$V\left(\phi_{\infty}\right)=2r$ for $r\rightarrow\infty$. Note that
$\mu=\frac{\mathcal{Q}}{\zeta-1}r_{\rm{H}}^{1-\zeta}$ is the chemical
potential. The blacken factor $f(r)$ recovers the standard JT black hole for
$f(r)=\frac{1}{l^{2}}\left(r^{2}-r_{\rm{H}}^{2}\right)$ as $a_{0}=Q^{2}=0$.
The Hawking temperature and the entropy of the black hole are given by

$T_{\rm{H}}=\frac{f^{\prime}\left(r_{\rm{H}}\right)}{4\pi}=\frac{2r_{\rm{H}}^{\zeta+1}+a_{0}r_{\rm{H}}^{\zeta-\eta}-Q^{2}}{4\pi
r_{\rm{H}}^{\zeta}},$ (2.4) $S=2\pi r_{\rm{H}}.$ (2.5)

This black hole solution is an asymptotically AdS black hole which shows a
clear van der Waals-Maxwell like phase structure. More discussions about this
black hole solution can be found in [62].

We consider a two-side asymptotically AdS black hole coupled with a bath. The
existence of the AdS boundary prevents black hole evaporation, so we apply the
transparent boundary condition to make the radiation enter the bath. The
asymptotically AdS black hole solution is (2.3a). As shown in figure 1, we
assume that the asymptotically AdS spacetime and the bath are in thermal
equilibrium. We denote $b_{\pm}\left(\pm t_{b},r_{b}\right)$ and
$a_{\pm}\left(\pm t_{a},r_{a}\right)$ as the locations of the cutoff surface
in the bath and the locations of the island in the asymptotically AdS
spacetime, respectively.

(a) without island

(b) with island

Figure 1: Penrose diagram for the black hole coupled to a thermal bath. The
bath can be viewed as a non-gravitational Minkowski spacetime where $a_{\pm}$
and $b_{\pm}$ correspond to the boundaries of the island and the cutoff
surface.

In order to obtain the maximal extension of the whole spacetime, we perform
the Kruskal transformation, defining the following coordinates,

$\displaystyle\text{ Right Wedge : }U$
$\displaystyle=-e^{-\kappa\left(t-r^{*}\right)},$ $\displaystyle
V=+e^{+\kappa\left(t+r^{*}\right)},$ (2.6) $\displaystyle\text{ Left Wedge :
}U$ $\displaystyle=+e^{-\kappa\left(t-r^{*}\right)},$ $\displaystyle
V=-e^{+\kappa\left(t+r^{*}\right)},$

and rewrite the metric as

$ds^{2}=-\Omega^{-2}dUdV.$ (2.7)

For the metric in (2.3a), the twist factor $\Omega$ is given by

$\Omega_{\text{2D}}=\frac{\kappa}{\sqrt{f(r)}}e^{\kappa r^{*}},$ (2.8)

where $\kappa=2\pi T_{\rm{H}}$ is the surface gravity, and $r^{*}=\int
f^{-1}(r)dr$ is the tortoise coordinate. For the flat bath region, $f(r)=1$,
$r^{*}=r$, so we can obtain the twist factor in the bath

$\Omega_{\text{bath}}=\kappa e^{\kappa r}.$ (2.9)

Now, we assume the system is in a vacuum state. So we can use the entanglement
entropy formula for 2D vacuum CFT [65], which reads as

$S_{\rm{vN}}=\frac{c}{6}\log
L=\frac{c}{6}\log\frac{\left(U_{1}-U_{2}\right)\left(V_{1}-V_{2}\right)}{\Omega_{1}\left(r_{1}\right)\Omega_{2}\left(r_{2}\right)},$
(2.10)

which represents the entanglement entropy from the contribution of the matter
field. Here $L$ is the geodesic distance between two boundary points of a
single interval, $\Omega_{1},\Omega_{2}$ are the twist factors for these two
boundary points, $U_{1,2}$ and $V_{1,2}$ are the Kruskal coordinates of these
two points in the spacetime. We will use this equation to calculate the
entanglement entropy in the following contexts.

##  3 Entanglement Entropy

In this section, we calculate the entanglement entropy of the radiation region
$\left[-\infty,b_{-}\right]\cup\left[b_{+},\infty\right]$ without the
contribution of the island at first. Considering the evolution of the pure
state, we only need to calculate the entanglement entropy of the region
$\left[b_{-},b_{+}\right]$ because of the complementarity principle, i.e. ,

$S_{\rm{rad}}\left(\text{no
island}\right)=S_{\rm{vN}}\left[b_{-},b_{+}\right]=\frac{c}{6}\log\frac{\left(U_{b_{+}}-U_{b_{-}}\right)\left(V_{b_{+}}-V_{b_{-}}\right)}{\Omega_{b_{+}}\Omega_{b_{-}}}.$
(3.1)

Inserting $b_{+}\left(t,r_{b}\right)$ and $b_{-}\left(-t,r_{b}\right)$ into
(2.6), we can obtain the Kruskal coordinates of $b_{+}$ and $b_{-}$. As shown
in figure 1(a), since coordinates $b_{+}$ and $b_{-}$ are in the bath, we can
obtain the twist factors by (2.9), i.e. ,
$\Omega_{b_{+}}=\Omega_{b_{-}}=\kappa e^{\kappa r_{b}}$. One inserts the
Kruskal coordinates of $b_{+}$ and $b_{-}$ into (3.1) and obtains

$\displaystyle S$
$\displaystyle=\frac{c}{6}\log\frac{\left(\mathrm{e}^{\kappa(\mathrm{t}+\mathrm{r_{b}})}+\mathrm{e}^{-\kappa(\mathrm{t}-\mathrm{r_{b}})}\right)^{2}}{\kappa^{2}e^{2\kappa
r_{b}}}$ (3.2) $\displaystyle=\frac{c}{3}\log\frac{2\mathrm{cosh}\left(\kappa
t\right)}{\kappa}.$

At late times, the approximation $\kappa t\gg 1$ is valid, so we have

$\lim_{t\to\infty}\cosh\left(\kappa t\right)\simeq\frac{1}{2}e^{\kappa t},$
(3.3)

and then obtain,

$S\simeq\frac{c}{3}\kappa t.$ (3.4)

Because of the late time approximation, we can find the increase of the
entanglement entropy agrees with the original Hawking’s result. The
entanglement entropy grows approximately linearly with time. Therefore, there
is a very sharp paradox here. The entanglement entropy diverging over time
violates the unitary principle of quantum mechanics. In order to satisfy the
unitary, we introduce the island to avoid such divergence.

###  3.1 Cutoff surface far away horizon

Now, we consider the cutoff surface that is far away from the horizon first
and then consider the cutoff surface that is near the horizon for calculating
the entanglement entropy with islands. The island emerges at late times as
shown in figure 1(b). At large distances and late times approximation, the
generalized entropy can be expressed as twice of the fine-grained entropy of
the region $\left[a_{+},b_{+}\right]$ with the contribution of the island
[66],

$S_{\rm{gen}}=4\pi
r_{a}+\frac{c}{3}\log\frac{\left(U_{b_{+}}-U_{a_{+}}\right)\left(V_{b_{+}}-V_{a_{+}}\right)}{\Omega_{b_{+}}\Omega_{a_{+}}},$
(3.5)

where the first term is the area term in two-dimensional gravity and the
second term represents the contribution of the matter field. $a_{+}$ and
$b_{+}$ are the locations of the island and the cutoff surface in the right
wedge of figure 1(b), respectively. One inserts the coordinates of
$b_{+}\left(t_{b},r_{b}\right)$ and $a_{+}\left(t_{a},r_{a}\right)$ into
(3.5), and obtains

$S_{\rm{gen}}=4\pi
r_{a}+\frac{c}{3}\log\frac{\sqrt{f(r_{a})}\left(2\mathrm{e}^{\kappa\left(r_{a}^{*}+\mathrm{r_{b}}\right)}\cosh\left[\kappa\left(t_{b}-t_{a}\right)\right]-\left(\mathrm{e}^{2\kappa\mathrm{r_{a}}^{*}}+\mathrm{e}^{2\kappa\mathrm{r_{b}}}\right)\right)}{\kappa^{2}e^{\kappa\left(r_{a}^{*}+r_{b}\right)}}.$
(3.6)

At first, we extremize $S_{\rm{gen}}$ with respect to $t_{a}$, i.e.

$\frac{\partial S_{\text{gen }}}{\partial
t_{a}}=\frac{c\kappa\sinh[\kappa(t_{b}-t_{a})]}{3(\cosh[\kappa(r_{a}^{*}-r_{b})]-\cosh[\kappa(t_{b}-t_{a})])}=0.$
(3.7)

The only solution is $t_{a}=t_{b}$. Then, (3.6) can be rewritten in the
following simpler form,

$S_{\rm{gen}}=4\pi r_{a}+\frac{2c}{3}\log\left[\frac{\mathrm{e}^{\kappa
r_{b}}-\mathrm{e}^{\kappa
r_{a}^{*}}}{\kappa}\right]+\frac{c}{6}\log\left[f\left(r_{a}\right)\mathrm{e}^{-2\kappa\left(r_{a}^{*}+r_{b}\right)}\right],$
(3.8)

where $r_{a},r_{b}$ are the radial-coordinates of the island and the cutoff
surface. To obtain the location of the island, we extremize $S_{\rm{gen}}$
with respect to $r_{a}$,

$\frac{\partial S_{\text{gen }}}{\partial
r_{a}}=4\pi-\frac{2c}{3}\frac{\kappa}{f\left(r_{a}\right)\left(\mathrm{e}^{\kappa\left(r_{b}-r_{a}^{*}\right)}-1\right)}+\frac{c}{6}\frac{f^{\prime}\left(r_{a}\right)-2\kappa}{f\left(r_{a}\right)}=0.$
(3.9)

Here, we take the near horizon approximation as follows,

$r^{*}\approx\frac{1}{2\kappa}\log\frac{r-r_{\rm{H}}}{r_{\rm{H}}},\quad
f(r)\approx 2\kappa\left(r-r_{\rm{H}}\right),$ (3.10)

where the denominator makes the whole formula dimensionless for the tortoise
coordinate. By the near horizon approximation, we insert the
$f\left(r_{a}\right)$ approximation into (3.9) and can be simplified as

$\frac{12\pi}{c}=\frac{e^{\kappa
r_{a}^{*}}}{\left(r_{a}-r_{\rm{H}}\right)\left(e^{\kappa r_{b}}-e^{\kappa
r_{a}^{*}}\right)}-\frac{1}{2}\frac{f^{\prime}\left(r_{a}\right)-f^{\prime}\left(r_{\rm{H}}\right)}{2\kappa\left(r_{a}-r_{\rm{H}}\right)}.$
(3.11)

Invoking $r_{a}^{*}$ approximation into (3.11), we obtain

$\displaystyle\frac{12\pi}{c}$
$\displaystyle=\frac{\sqrt{\frac{r_{a}-r_{\rm{H}}}{r_{\rm{H}}}}}{\left(r_{a}-r_{\rm{H}}\right)\left(e^{\kappa
r_{b}}-\sqrt{\frac{r_{a}-r_{\rm{H}}}{r_{\rm{H}}}}\right)}-\frac{1}{2}\frac{f^{\prime\prime}\left(r_{\rm{H}}\right)}{2\kappa}$
(3.12)
$\displaystyle\simeq\frac{1}{\sqrt{r_{\rm{H}}\left(-r_{\rm{H}}+r_{a}\right)}\left(e^{\kappa
r_{b}}-\sqrt{\frac{-r_{\rm{H}}+r_{a}}{r_{\rm{H}}}}\right)},$

By solving (3.12), we can obtain the location of islands

$r_{a}=r_{\rm{H}}-\frac{c}{12\pi}+\frac{1}{2}r_{\rm{H}}e^{2\kappa
r_{b}}-\frac{1}{2}\sqrt{r_{\rm{H}}}e^{\kappa
r_{b}}\sqrt{-\frac{c}{3\pi}+r_{\rm{H}}e^{2\kappa r_{b}}}.$ (3.13)

Simplifying the function $r_{a}\left(c\right)$ by Taylor’s expansion in series
of $c$, we obtain

$r_{a}=r_{\rm{H}}+\frac{c^{2}}{144\pi^{2}r_{\rm{H}}}e^{-2\kappa r_{b}}+{\cal
O}\left(c^{3}\right).$ (3.14)

From (3.14), we can see that the location of the boundary of the island is
outside the event horizon. We can insert (3.14) into (3.8) and omit the high
order terms. We can obtain the expression of the entanglement entropy,

$\displaystyle S$ $\displaystyle\simeq 4\pi
r_{\rm{H}}+\frac{c}{6}\log\frac{2r_{\rm{H}}}{\kappa^{3}}+{\cal
O}\left(c^{2}\right)$ (3.15) $\displaystyle\simeq 2S_{BH}.$

Because of the emergence of islands, the entanglement entropy does not
increase but arrives at a maximal value which is twice the Bekenstein-Hawking
entropy. The logarithm term comes from the quantum effect of the matter field.
Therefore, for the whole evaporation process, the entanglement entropy
increases approximately linearly at the beginning, and then does not increase
after reaching the maximum value.

###  3.2 Cutoff surface near horizon

Now let us discuss the case where the cutoff surface is near the horizon,
which is different from the case where the cutoff surface is far from the
black hole horizon. We follow the similar logic in [25]. When it nears the
horizon, the generalized entropy is given by

$S_{\rm{gen}}\simeq 4\pi r_{a}-2Kc\frac{8\pi r_{b}}{L^{2}},$ (3.16)

where $K$ is a constant and $L$ is the geodesic distance between the endpoint
of the island and the cutoff surface,

$\displaystyle L$
$\displaystyle\simeq\frac{1}{\sqrt{2\kappa}}\int_{r_{a}}^{r_{b}}\frac{dr}{\sqrt{r-r_{\mathrm{H}}}}$
(3.17)
$\displaystyle\simeq\frac{\sqrt{2}}{\sqrt{\kappa}}\left(\sqrt{r_{b}-r_{\rm{H}}}-\sqrt{r_{a}-r_{\rm{H}}}\right).$

We substitute (3.17) into (3.16) and extremize $S_{\rm{gen}}$ with respect to
$r_{a}$,

$\frac{\partial S_{\text{gen }}}{\partial r_{a}}=\frac{2r_{b}c\kappa
K}{\left(\sqrt{r_{b}-r_{\rm{H}}}-\sqrt{r_{a}-r_{\rm{H}}}\right)^{3}\sqrt{r_{a}-r_{\rm{H}}}}-1=0.$
(3.18)

where approximations $\sqrt{r_{a}-r_{\rm{H}}}\ll 1$ and $r_{b}\simeq
r_{\rm{H}}$ are taken into account. The minimization occurs at a smaller
solution, satisfying $\sqrt{r_{a}-r_{\rm{H}}}\ll\sqrt{r_{b}-r_{\rm{H}}}$, we
obtain the location of the island as

$r_{a}=r_{\rm{H}}+\frac{\left(2r_{\rm{H}}c\kappa
K\right)^{2}}{\left(r_{b}-r_{\rm{H}}\right)^{3}}.$ (3.19)

Obviously, the location of the boundary of the island is also outside and very
close to the event horizon. The result is similar to the situation that the
cutoff surface is far from the horizon (3.14).

Inserting (3.19) to (3.16), we can obtain the entanglement entropy as

$\displaystyle S$ $\displaystyle\simeq 4\pi r_{\rm{H}}-8\pi c\kappa
K\frac{r_{\rm{H}}}{r_{b}-r_{\rm{H}}}$ (3.20) $\displaystyle\simeq
2S_{\rm{BH}}.$

Again, the first term of the entanglement entropy is twice of the Bekenstein-
Hawking entropy. The second term is a small number and can be omitted. Because
of the emergence of the island, whatever we choose the cutoff surface, the
entanglement entropy is about twice the Bekenstein-Hawking entropy.

In summary, the entanglement entropy without islands linear growth diverges in
later times leading to the information paradox. However, once the island is
considered, the configuration of space-time changes: a region that is not
connected with the external radiation region appears inside the black hole,
i.e., the entanglement wedge of radiation contains the black hole interior. At
last, the entanglement entropy reaches a constant at late times with island.
According to the island paradigm, we obtain the unitary result of the
evolution of the entanglement entropy.

##  4 Page curve and Page time

In this section, we obtain the change of the entanglement entropy of the
evolution of the black hole. At the early time, from (3.4), the entanglement
entropy linearly increases. But because of the island paradigm, after the Page
time, the emergence of the island leads to the entanglement entropy does not
increase but is a constant (3.15). Therefore we can obtain the Page curve
naturally in figure 2. By considering (3.4) and (3.15), we obtain the Page
time easily,

$t_{\text{Page }}=\frac{12\pi r_{\rm{H}}}{c\kappa}.$ (4.1)

Figure 2: Page curve. The entanglement entropy linearly increases at the early
time. Then after the Page time, the entanglement entropy arrives at the
maximum value – twice of the Bekenstein-Hawking entropy.

Also, we can obtain the scrambling time which is defined as the minimum time
interval in which the information is recovered [68]. In the island paradigm,
this corresponds to the period from the cutoff surface to the boundary of the
island. We assume the time interval between sending the information from the
cutoff surface $r=r_{b}$ to the black hole and recovering on the island
surface $r=r_{a}$ to be

$\Delta
t=r^{*}_{b}-r^{*}_{a}\simeq\frac{1}{2\kappa}\log\frac{r_{b}-r_{\rm{H}}}{r_{a}-r_{\rm{H}}}.$
(4.2)

Substituting the value of $r_{a}$ from equation (3.14) in equation (4.2), the
scrambling time is given by

$t_{\rm{scr}}=\frac{1}{2\kappa}\log\frac{144\pi^{2}\left(r_{b}-r_{\rm{H}}\right)r_{\rm{H}}}{c^{2}e^{-2\kappa
r_{b}}}\simeq\frac{1}{2\pi T_{\rm{H}}}\log S_{\rm{BH}},$ (4.3)

where we set $r_{b}$ to have the same order with $r_{\rm{H}}$, $T_{\rm{H}}$ is
the Hawking temperature defined (2.4), and $S_{\rm{BH}}=2\pi r_{\rm{H}}$ is
the Bekenstein-Hawking entropy. This result is consistant with [25, 67].

##  5 Phase transition of the black hole

So far, we calculate the change of the entanglement entropy of the whole
evaporation process for the special black hole solution and obtain the
location of the island and the scrambling time. According to our calculation
results, we plot the unitary Page curve. Now we consider the phase transition
and analyze the relationship between the Page curve and the phase structure.
We consider the charged black hole solution in canonical ensemble with
$\eta=1,\zeta=2$, and set the AdS radius $l=1$. Then (2.3c) can be simplified
as

$f(r)=r^{2}-r^{2}_{\rm{H}}+a_{0}\ln\frac{r}{r_{\rm{H}}}+Q^{2}\left(r^{-1}-r^{-1}_{\rm{H}}\right).$
(5.1)

For this charged black hole, we can obtain the Hawking temperature and the
entropy from (2.4) and (2.5), which read as

$T_{\rm{H}}=\frac{2r_{\rm{H}}^{3}+a_{0}r_{\rm{H}}-Q^{2}}{4\pi
r_{\rm{H}}^{2}},\quad S=2\pi r_{\rm{H}}.$ (5.2)

We also can obtain the expression of the free energy of this black hole
solution [62],

$F=-\frac{1}{2}r_{\rm{H}}^{2}-\frac{Q^{2}}{2}r_{\rm{H}}^{-1}-\frac{a_{0}}{2}+\frac{a_{0}}{2}\ln
r_{\rm{H}}.$ (5.3)

In the following, we study the chargeless case first and then consider the
charged black hole.

###  5.1 $Q=0$

For the special case with $Q=0$, the black hole thermodynamics phase diagram
reduces to the Hawking-Page phase transition. We can write down the Hawking
temperature and the free energy as

$T_{\rm{H}}=\frac{r_{\rm{H}}}{2\pi}+\frac{a_{0}}{4\pi r_{\rm{H}}},\quad
F=-\frac{1}{2}r_{\rm{H}}^{2}-\frac{a_{0}}{2}+\frac{a_{0}}{2}\ln r_{\rm{H}}.$
(5.4)

At the same temperature, for the small black hole, its horizon radius is

$r_{s}=\frac{1}{2}\left(2\pi
T_{\rm{H}}-\sqrt{2}\sqrt{-a_{0}+2\pi^{2}T_{\rm{H}}^{2}}\right),$ (5.5)

while, for the large black hole, its horizon radius is

$r_{l}=\frac{1}{2}\left(2\pi
T_{\rm{H}}+\sqrt{2}\sqrt{-a_{0}+2\pi^{2}T_{\rm{H}}^{2}}\right).$ (5.6)

Obviously, there is a minimum temperature
$T_{\rm{H}\left(\rm{min}\right)}=\frac{\sqrt{a_{0}}}{\sqrt{2}\pi}$. For the
minimum temperature, two radii are coincident,
$r_{l}=r_{s}=\frac{\sqrt{a_{0}}}{\sqrt{2}}$. Moreover, there are no black hole
phases below the minimum temperature.

We can plot the $F$-$T_{\rm{H}}$ diagram as follows.

Figure 3: The free energy $F$ as a function of temperature $T_{\rm{H}}$ at
$a_{0}=0.5,1,2$. The situation of the Hawking-Page phase structure. The low
and high branches correspond to the large and small black holes.

As shown in figure 3, the free energy of the large black hole is always lower
than the free energy of the small black hole. Therefore, for the same
temperature, the large black hole is the stable phase. Considering the
Hawking-Page phase transition, the small black hole is thermodynamically
unstable. Therefore, we do not need to consider the island and the Page time
of the small black hole, including the extremal black hole.

To study the effect on the Page curve from the Hawking-Page phase transition,
we consider the situation in the absence of the islands first. From equation
(3.4), the entanglement entropy increases linearly at late times. As shown in
figure 4, we obtain the figure for the increased rate of the entanglement
entropy and the Page time,

$\frac{S}{t}=\frac{c}{3}\kappa.$ (5.7)

We choose $\eta=1,\zeta=2,Q=0,c=1.5$ for (4.1) and (4.2), then we obtain

$t_{\text{Page }}=\frac{16\pi}{2+a_{0}r_{\rm{H}}^{-2}},\quad
t_{\text{scr}}=\frac{2r_{\rm{H}}}{2r_{\rm{H}}^{2}+a_{0}}\log 2\pi r_{\rm{H}}.$
(5.8)

Figure 4: The increased rate of the entanglement entropy and the Page time as
a function for the radius at $a_{0}=0.5,1,2$. The dotted line represents the
unstable small black hole.

Now we consider the island paradigm,

$S_{\mathrm{rad}}\left(r_{\mathrm{H}},t\right)=\text{Min}\left[S\left(\text{no
island}\right),S\left(\text{with
island}\right)\right]=\text{Min}\left[\frac{c}{3}\log\frac{2\cosh(\kappa
t)}{\kappa},2S_{\mathrm{BH}}\right].$ (5.9)

The whole process of the evolution of the entanglement entropy should satisfy
(5.9). Considering the Hawking-Page phase transition, we set $c=1.5$ and plot
the Page curve as a function of time $t$ and radius $r_{\rm{H}}$ for $a_{0}=1$
in figure 6.

Figure 5: The scrambling time as a function for the radius at $a_{0}=0.5,1,2$.
The dotted line represents the unstable small black hole. Figure 6: Page
curves as a function of time $t$ and radius $r_{\rm{H}}$ for $a_{0}=1$ and
$c=1.5$.

###  5.2 $Q\neq 0$

For the general case, we set $\eta=1,\zeta=2,a_{0}=1,l=1$ and obtain the
Hawking temperature and the free energy as

$T_{\mathrm{H}}=\frac{2r_{\mathrm{H}}^{3}+r_{\mathrm{H}}-Q^{2}}{4\pi
r_{\mathrm{H}}^{2}},\quad
F=-\frac{1}{2}r_{\mathrm{H}}^{2}-\frac{Q^{2}}{2}r_{\mathrm{H}}^{-1}-\frac{1}{2}+\frac{1}{2}\ln
r_{\mathrm{H}}.$ (5.10)

By extremizing $T_{\rm{H}}$ with respect to $r_{\rm{H}}$,

$\displaystyle\frac{\partial T_{\rm{H}}}{\partial
r_{\rm{H}}}=2r_{\rm{H}}^{3}-r_{\rm{H}}+2Q^{2}$ $\displaystyle=0,$ (5.11)

we can find only two real, positive solutions obey (5.11). These two solutions
coalesce at the critical point,

$\displaystyle\frac{\partial^{2}T_{\rm{H}}}{\partial^{2}r_{\rm{H}}}=r_{\rm{H}}-3Q^{2}=0.$
(5.12)

According to equations (5.11) and (5.12), we can obtain
$r_{\rm{H}\left(\text{crit}\right)}=\frac{1}{\sqrt{6}}$,
$Q_{\left(\text{crit}\right)}=\frac{1}{\sqrt{3\sqrt{6}}}$,
$T_{\rm{H}\left(\text{crit}\right)}=\frac{1}{2\pi}\sqrt{\frac{3}{2}}$. Also,
we can plot the $F$-$T_{\rm{H}}$ diagram for the charged situation as follows.

Figure 7: (a): The free energy $F$ as a function of temperature $T_{\rm{H}}$
at $Q^{2}=0.04,0.06$. (b): The free energy $F$ as a function of temperature
$T_{\rm{H}}$ at $Q^{2}=Q^{2}_{\text{ crit }},0.5,1$.

There is a section of the well-known “swallowtail” shape in figure 7. The
intersection points correspond to the phase transition point in the fixed
charge ensemble for $Q^{2}<Q^{2}_{\rm{crit}}$. The most stable phase requires
the lowest free energy. When $Q^{2}>Q^{2}_{\rm{crit}}$, the swallowtail
structure disappears in figure 7, so the phase transition also does not
happen.

To investigate the phase transition effects on the Page curve, we plot the
increased rate of the entanglement entropy as a function of the radius for
several different values of $Q^{2}$,

$\frac{S}{t}=\frac{c}{3}\kappa=\frac{2r_{\rm{H}}^{3}+r_{\rm{H}}-Q^{2}}{2r_{\rm{H}}^{2}}.$
(5.13)

According to (4.1) and (4.3), we set $c$=1.5 and write down the Page time and
the scrambling time versus the radius for any $Q^{2}$,

$\displaystyle t_{\text{Page }}$
$\displaystyle=\frac{16\pi}{2+r_{\rm{H}}^{-2}-Q^{2}r_{\rm{H}}^{-3}},$ (5.14)
$\displaystyle t_{\text{scr }}$
$\displaystyle=\frac{2r_{\rm{H}}^{2}}{2r_{\rm{H}}^{3}+r_{\rm{H}}-Q^{2}}\log
2\pi r_{\rm{H}}.$

Figure 8: The increased rate of the entanglement entropy and the Page time as
a function for the radius at $Q^{2}=0.04,0.06,\frac{1}{3\sqrt{6}},0.5,1$.

Obviously, in figure 8, the increased rate of the entanglement entropy is not
a monotonic function of the radius when $Q^{2}<Q^{2}_{\rm{crit}}$. The same
increased rate corresponds to three different radius values which represent
the “small, middle, large” black hole. However, in real physical processes,
the lower the free energy is, the more stable phase is. As figure 7, the
intersection points are just the critical points where the phase transition
happens. Above the critical temperature, the large black hole is more stable,
but below the critical temperature, the small black hole is more stable. Thus,
figure 7 behaves like the van der Waals-Maxwell phase structure.

In figure 8, it is obvious that the Page time will be a constant
$\left(t_{\rm{Page}}=8\pi\right)$, when $r_{\rm{H}}$ approaches infinity. For
the large black hole, the charge and entropy do not influence the Page time.
However, for the extremal black hole, the Page time is divergent. Since the
middle black hole is unstable, so that the physical meaning of the Page time
of the middle black hole is not clear when $Q^{2}<Q^{2}_{\rm{crit}}$.

Figure 9: The scrambling time as a function for the radius at
$Q^{2}=0.04,0.06,\frac{1}{3\sqrt{6}},0.5,1$.

Considering the island paradigm (5.9), we set $c=1.5$ and plot the Page curve
of a series of small and large black holes at $Q^{2}=0.04$ in figure 10. As
shown in figure 10, for the small black hole, the Page time approaches zero.
It is different from figure 8. Because we take the late time approximation in
figure 8, it is inaccurate for the Page time of the small black hole. But we
do not use the late time approximation in (5.9), therefore the result of
figure 10 is reasonable.

Figure 10: Page curves for different black holes with $c=1.5$. (a): Page
curves as a function of time $t$ and radius $r_{\rm{H}}$ for a series of the
small black hole at $Q^{2}=0.04$. (b): Page curves as a function of time $t$
and radius $r_{\rm{H}}$ for the large black hole at $Q^{2}=0.04$.

Figure 11: Page curves for different temperatures. (a): Page curve at the
temperature which is higher than the critical temperature. (b): Page curve at
the temperature which is lower than the critical temperature.

In figure 11, we plot the Page curve of the black hole with the special phase
structure. Above the critical temperature, we can only obtain the Page curve
of the large black hole. However, below the critical temperature, we can only
obtain the Page curve of the small black hole. The lower the temperature, the
greater the Page time. This implies that the lower the temperature, the later
the island appears in the black hole’s interior.

Figure 12: Page curve for the critical temperature. The large and small black
holes coexist.

Taking into account the critical temperature, because of the instability of
the middle black hole, the range of the radius of the middle black hole would
not become the black hole phase, so the island does not emerge. As the phase
transition of the black hole, the location of the island also has a jump. In
figure 12, when the black hole temperature is the critical temperature, the
large and small black holes coexist.

Finally, we consider the different situations without phase transition when
$Q^{2}>Q^{2}_{\rm{crit}}$. The green branch corresponds to the critical charge
in figure 8. As shown in figure 8, we know the increased rate of the
entanglement entropy beyond the critical charge is the monotone function of
the radius. As shown in figure 8, with the increase of the charge, the Page
time is later. There is no more phase transition, so the Page curve has no
special situation for any size black hole just like in figure 2. We set
$c=1.5$ and plot the Page curve as a function of time $t$ and radius
$r_{\rm{H}}$ at $Q^{2}=0.5$ in figure 13.

Figure 13: Page curves as a function of time $t$ and radius $r_{\rm{H}}$ at
$Q^{2}=0.5$ and $c=1.5$.

##  6 Discussion and Conclusion

In summary, we consider the information paradox for a special black hole
derived from the deformed JT gravity with the van der Waals-Maxwell like phase
structure. To make the black hole evaporate, we construct the eternal black
hole by gluing the flat bath. Then we use the island formula to calculate the
entanglement entropy. At early time, the entanglement entropy linearly
increases without islands. It matches Hawking’s original calculations. At
later times, because of the island, the entanglement entropy reaches a
constant value which is twice of the Bekenstein-Hawking entropy. Finally, we
obtain the scrambling time and the Page curve. We obtain the unitary result
and the island outside the event horizon that is consistent with the result of
the eternal black hole. Moreover, we consider different situations for the
cutoff surface and find the final result is independent of the distance of the
cutoff surface.

Moreover, we consider the impact of the phase transition. Phase transition is
an important and interesting thermodynamic phenomenon. In fixed charged
ensembles with different charges, the black hole has different phase
transitions. There is the Hawking-Page phase transition in the black hole
without charges. As the charge increases, There is the van der Waals-Maxwell-
like phase transition. But if the charge is beyond the critical charge, there
are no phase transitions. We study the island and the Page time for the black
hole with these phase structures to obtain some results about the influence
yielded on the phase transition.

For an neutral black hole without charges, there is only the Hawking-Page
phase transition. For the same temperature, there are two black hole phases –
the small and large black holes. By calculating the free energy and the
Hawking temperature, we obtain figure 3, where the large black hole with lower
free energy is more stable. Note that the system has a minimum temperature for
forming the black hole. So it is meaningless to discuss the island and the
Page time below this minimum temperature in this situation. Thus, we only can
obtain the Page curve of the large black hole.

For charged black holes, at first, we consider $Q<Q_{\rm{crit}}$, the black
hole would have the van der Waals-Maxwell-like phase structure that three
different black hole phases may correspond to the same temperature. By
calculating the free energy and the Hawking temperature, we obtain figure 7,
the lowest free energy phase is stable, so only one of the three phases is the
stable one. There is a critical temperature where the small black hole phase
would jump to the large black hole phase. Of course, the Page time about the
middle black hole is meaningless. The situations with the temperature beyond,
below, and equal to the critical temperature respectively correspond to figure
11 which the large black hole is more stable, figure 11 which the small black
hole is more stable, and figure 12 which the small and large black holes
coexist. Then we consider $Q\geq Q_{\rm{crit}}$ case in which is the no phase
transition happens. The Page time for any temperature is well defined in this
situation. Different from the no phase transition situation, the Page time of
the small black hole approaches zero in figure 10. We do not use the late time
approximation to calculate the Page time in (5.9). The scrambling time curve
monotonically decreases in the no phase transition situation. But in the phase
transition situation, there is a maximum scrambling time.

In the doubly-holographic system [69], one simply includes a second Karch-
Randall brane as a gravitating bath. One finds if the bath itself gravitates,
the original Page curve vanishes. And the quantum extremal surface is just at
the horizon. These successful works about the introduction of the gravitating
bath are in the high dimension [69, 31, 70]. But there is no graviton in two
dimensions. So we can not introduce the KR brane as the gravitating bath in
our 2d gravity model. We will consider the reliable construction of the
gravitating bath in future work.

In conclusion, we use a simple black hole model with a rich phase structure to
compute and analyze the Page curve and study the effect of the phase
transition to Page curves. We find the important relationship between the
phase transition and the black hole information paradox. We hope that the deep
understanding and keen insight of the 2D gravity model in this paper can be
applied to the higher-dimensional space-time we live in. As an important
phenomenon, the phase transition might be interesting for understanding the
information paradox.

## Acknowledgement

We would like to thank Cheng Ran, Hai-Ming Yuan, Yu-Qi Lei and Qing-Bin Wang
for helpful discussions. The study was partially supported by NSFC, China
(grant No.11875184 and No.12275166).

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# Superconductivity near a nematoelastic quantum critical point

Vanuildo S. de Carvalho  Instituto de Física, Universidade Federal de Goiás,
74.001-970, Goiânia-GO, Brazil  Hermann Freire  Instituto de Física,
Universidade Federal de Goiás, 74.001-970, Goiânia-GO, Brazil

###### Abstract

We study the pairing instability of a two-dimensional metallic system induced
by Ising-nematic quantum fluctuations in the presence of an unavoidable
relevant coupling of the nematic order parameter to the elastic modes
(acoustic phonons) of the lattice. We find that this nematoelastic coupling
$\lambda_{\mathrm{latt}}$ leads to a decrease of both the superconducting (SC)
critical temperature $T_{c}$ and the gap function $\Delta$, regardless of the
gap symmetry. Interestingly, we show that $\lambda_{\mathrm{latt}}$ provides a
knob that allows us to investigate the emergence of the SC phase at low
temperatures, as an instability from either a non-Fermi liquid or a Fermi
liquid normal state. The phase transitions between the SC and these normal
states are characterized by different critical exponents, which may also vary
for each gap symmetry. Finally, we argue that these results might explain the
dependence of $T_{c}$ in the vicinity of the nematic quantum critical point
exhibited by the compound FeSe1-xSx.

_Introduction.–_ A theoretically controlled analysis of two-dimensional metals
near an Ising-nematic quantum critical point (QCP) remains a big challenge
nowadays in the field of strongly correlated systems. It is now well-
established that the renormalization group approach with large-$N$ expansion
eventually breaks down for this theory at low enough energies Lee (2009);
Metlitski and Sachdev (2010a); *Metlitski-PRBb(2010); Mross  _et al._ (2010);
Lee (2018). The Ising-nematic phase refers to a long-range instability with
$\mathbf{Q}=0$ momentum transfer that leads to the lowering of the point-group
rotational symmetry, while preserving the properties of the underlying system
with respect to translational symmetry Fradkin  _et al._ (2010); Fernandes
_et al._ (2014). Recently, both analytical and numerical approaches Lee
(2009); Metlitski and Sachdev (2010a); Mross  _et al._ (2010); Hartnoll  _et
al._ (2014); Metlitski  _et al._ (2015); Lederer  _et al._ (2015); Schattner
_et al._ (2016); Lederer  _et al._ (2017); Lee (2018); Berg  _et al._ (2019)
have shown that the emergence of such a phase could potentially explain some
of the physical properties observed in the phase diagram of high-temperature
superconductors, such as, e.g., the cuprates Hinkov  _et al._ (2008); Daou
_et al._ (2010); Sato  _et al._ (2017), the iron-based superconductors Chu
_et al._ (2010); Kasahara  _et al._ (2012); Hosoi  _et al._ (2016);
Licciardello  _et al._ (2019); Reiss  _et al._ (2020), and also in other
correlated materials Ronning  _et al._ (2017); Cao  _et al._ (2021).

Although the qualitative comparison between the theoretical predictions with
experimental data in these systems is reasonably good, a clear picture
regarding the impact of electronic nematicity on the corresponding phase
diagrams is still lacking. This occurs because for most systems this
instability appears along with the emergence of other long-range orders such
as antiferromagnetism and superconductivity, which complicate the analysis. In
this regard, it was recently found that the iron-based superconductors
FeSe1-xSx Hosoi  _et al._ (2016) and LaFeAsO1-xFx Yang  _et al._ (2015)
display a quantum phase transition (QPT) to an electronic nematic state
disentangled from other magnetic or charge ordered phases, and thus they
constitute the most promising platform to study the effects of electronic
nematicity.

However, a more realistic attempt to investigate such a quantum critical
theory should also include the unavoidable interaction of the nematic order-
parameter with the elastic modes (acoustic phonons) of the lattice Cowley
(1976); Qi and Xu (2009); Fernandes  _et al._ (2010); Cano  _et al._ (2010);
Labat and Paul (2017); Karahasanovic and Schmalian (2016). By taking into
account this relevant interaction, previous works have predicted that the low-
temperature thermodynamic Paul and Garst (2017) and transport de Carvalho and
Fernandes (2019); Wang and Berg (2019); Vieira  _et al._ (2020) properties of
the corresponding quantum critical state have a tendency of becoming Fermi-
liquid-like below an energy scale set by the nematoelastic interaction. This
happens because the shear strain associated with this interaction constrains
the quantum critical region to a few high symmetry directions Labat and Paul
(2017). Remarkably, the authors of Refs. Reiss  _et al._ (2020); Bristow  _et
al._ (2020) have shown that in the vicinity of the nematic QCP, the electrical
resistivity clearly transitions from a non-Fermi liquid (NFL) to a Fermi
liquid (FL) regime as the temperature is lowered, in good agreement with the
theoretical results that consider the effects of the nematoelastic coupling de
Carvalho and Fernandes (2019); Wang and Berg (2019); Vieira  _et al._ (2020).

Figure 1: Phase diagram of a 2D metallic system at the onset of Ising-
nematicity when the nematic degrees of freedom couple to the acoustic phonons
by means of an effective nematoelastic interaction
$\lambda_{\mathrm{latt}}\sim g^{2}_{\mathrm{latt}}$. In the regime where
$\lambda_{\mathrm{latt}}$ is negligible compared to the effective electron
interaction $\gamma\sim g^{2}$, the SC state emerges out of a non-Fermi liquid
(NFL). On the other hand, as $\lambda_{\mathrm{latt}}$ approaches a lattice-
induced quantum critical point (QCP∗), whose location depends on $\gamma$ as a
power-law, the normal state from which SC emerges is a Fermi liquid (FL). The
solid line denoted by $T_{c}$ refers to the transition to the SC state
calculated in the present work [see Eq. (11)]. The dotted-dashed line
represents a crossover between the FL and the NFL regimes, which is given by
$T_{\mathrm{FL}}\sim\lambda^{3/2}_{\mathrm{latt}}E_{F}$ Paul and Garst (2017),
where $E_{F}$ is the Fermi energy.

In this Letter, we investigate the impact of the nematoelastic coupling,
$g_{\mathrm{latt}}$, on a two-dimensional (2D) SC state that emerges near the
onset of an Ising-nematic order. In order to do that, we apply the strong-
coupling Eliashberg theory to obtain the critical SC temperature $T_{c}$ and
the gap function $\Delta(\theta,\omega_{n})$ around a circular Fermi surface
at $T=T_{c}$, by considering both _s_ -wave and _d_ -wave symmetries. In this
connection, we highlight two recent works that address similar issues
regarding this problem, which can be viewed as complementary to our present
study (see Refs. Labat and Paul (2017); Klein and Chubukov (2018); *Klein-
NPJ(2019)). The first one Labat and Paul (2017) describes, within a weak-
coupling BCS theory, how SC emerges near a nematic QCP for a system in the
presence of such a nematoelastic interaction. The authors find that $T_{c}$
only increases if the nematoelastic coupling is strong enough and the system
is dominated by non-nematic interactions Labat and Paul (2017). By contrast,
the second work Klein and Chubukov (2018) focus on the properties of the SC
state driven by strong nematic fluctuations (but without the nematoelastic
coupling) from a Eliashberg-theory perspective. As a result, they find that
nematic fluctuations always favor an _s_ -wave SC state, despite it becoming
almost degenerate with a _d_ -wave one in the weak-coupling regime Klein and
Chubukov (2018).

Our main results are summarized in the phase diagram of Fig. 1. The increase
of $\lambda_{\mathrm{latt}}\sim g^{2}_{\mathrm{latt}}$ is always detrimental
to superconductivity and, therefore, reduces $T_{c}$ and the
$\Delta(\theta,\omega_{n})$. This behavior is qualitatively similar for both
_s_ -wave and _d_ -wave pairing, although the reduction in $T_{c}$ and
$\Delta(\theta,\omega_{n})$ becomes stronger in the latter case. In the limit
where $\lambda_{\mathrm{latt}}$ is in the vicinity of a lattice-induced
quantum critical point (QCP∗), whose location depends on the effective
electron interaction $\gamma\sim g^{2}$ as a power-law, the system undergoes a
SC-FL quantum phase transition. In addition, the effective coupling
$\lambda_{\mathrm{latt}}$ allows us to obtain SC states emerging from
different metallic states, represented by the FL and NFL phases. This leads to
interesting new predictions, because the NFL-SC and the FL-SC phase
transitions turn out to be described in terms of different critical exponents,
which may also vary for each gap symmetry. As a result, we argue that the
suppression of $T_{c}$ in the vicinity of the nematic QCP for the compound
FeSe1-xSx Reiss  _et al._ (2020) may be interpreted as a fingerprint of
nematoelastic quantum criticality.

_Model.–_ We model the instability to the nematic state in a 2D system in
terms of the coupling of the nematic order parameter to the electronic
quasiparticles Lee (2009); Metlitski and Sachdev (2010a). We include the
influence of the lattice on that order parameter through a linear coupling to
the local orthorhombic strain
$\epsilon(\mathbf{r})\equiv\epsilon_{xx}(\mathbf{r})-\epsilon_{yy}(\mathbf{r})$
Qi and Xu (2009); Fernandes  _et al._ (2010); Cano  _et al._ (2010); Labat and
Paul (2017); Karahasanovic and Schmalian (2016), where the strain is defined
in terms of the displacement vector $\bm{u}(\mathbf{r})$ by means of
$\epsilon_{ij}=\partial_{i}u_{j}+\partial_{j}u_{i}$ Landau and Lifshitz
(1970); Cowley (1976). The minimal Hamiltonian describing this system reads

$\displaystyle H$ $\displaystyle=H_{\text{n-elec}}+H_{\text{n-latt}},$ (1)
$\displaystyle H_{\text{n-elec}}$
$\displaystyle=\sum\limits_{\mathbf{k},\sigma}\xi_{\mathbf{k}}\psi^{\dagger}_{\sigma}(\mathbf{k})\psi_{\sigma}(\mathbf{k})+\sum_{\mathbf{q}}\Omega(\mathbf{q})\varphi(\mathbf{q})\varphi(-\mathbf{q})$
$\displaystyle+g\sum\limits_{\mathbf{q},\mathbf{k},\sigma}f(\mathbf{k})\psi^{\dagger}_{\sigma}\left(\mathbf{k}-\frac{\mathbf{q}}{2}\right)\psi_{\sigma}\left(\mathbf{k}+\frac{\mathbf{q}}{2}\right)\varphi(\mathbf{q}),$
(2) $\displaystyle H_{\text{n-latt}}$
$\displaystyle=\sum\limits_{\mathbf{q}\neq
0}\left[\frac{1}{2}\bm{u}^{\dagger}_{\mathbf{q}}\bm{\mathcal{N}}(\mathbf{q})\bm{u}_{\mathbf{q}}+ig_{\mathrm{latt}}\mathbf{a}_{\mathbf{q}}\cdot\bm{u}_{\mathbf{q}}\varphi(-\mathbf{q})\right],$
(3)

where $\psi^{\dagger}_{\sigma}(\mathbf{k})$ ($\psi_{\sigma}(\mathbf{k})$) is
the creation (annihilation) operator for fermions with spin projection
$\sigma\in\\{\uparrow,\downarrow\\}$ and band dispersion $\xi_{\mathbf{k}}$,
$\Omega_{\mathbf{q}}$ is the dispersion of the nematic bosons described by
$\varphi(\mathbf{q})$, $g$ and $g_{\mathrm{latt}}$ are, respectively, the
nematic and the nematoelastic interactions, and $f(\mathbf{k})\equiv(\cos
k_{x}-\cos k_{y})$ denotes a _d_ -wave form factor for a nematic state with
$B_{1g}$ symmetry. Moreover, the acoustic phonons are described in terms of
the Fourier-transformed displacement vector $\bm{u}_{\mathbf{q}}$,
$\mathbf{a}_{\mathbf{q}}=(q_{x},-q_{y},0)$ is a 2D vector, and
$\bm{\mathcal{N}}(\mathbf{q})$ stands for the matrix of the elastic constants
$C_{ij}$ Landau and Lifshitz (1970); Cowley (1976), which for a 2D system with
tetragonal symmetry is given by

$\bm{\mathcal{N}}(\mathbf{q})=\begin{pmatrix}C_{11}q_{x}^{2}+C_{66}q_{y}^{2}&(C_{12}+C_{66})q_{x}q_{y}\\\
(C_{12}+C_{66})q_{x}q_{y}&C_{66}q_{x}^{2}+C_{11}q_{y}^{2}\end{pmatrix}.$ (4)

Figure 2: (a)-(b) Behavior of the SC gap (in units of the Fermi energy) with
_s_ -wave symmetry along the Fermi surface and for Matsubara frequencies
$\omega_{n}=(2n+1)\pi T$ obtained by setting $\gamma=0.40$ and then solving
the linearized Eliashberg equations at $T=T_{c}$ for (a)
$\lambda_{\mathrm{latt}}=0$ and (b) $\lambda_{\mathrm{latt}}=0.10$. Notice
that the effect of increasing $\lambda_{\mathrm{latt}}$ is to diminish
$\bar{\Delta}^{(s)}_{n}(\theta)$ along the whole Fermi surface. (c) The SC gap
at the hot spot $\theta=0$ decreases monotonically with
$\lambda_{\mathrm{latt}}$. At
$\lambda_{\mathrm{latt}}=\lambda_{\mathrm{latt},c}$, which is given here by
$\lambda_{\mathrm{latt},c}\approx 0.21$, the gap
$\bar{\Delta}^{(s)}_{n}(\theta=0)$ becomes degenerate in both frequency and
momentum. Panels (d)-(f) show the behavior of a SC gap with _d_ -wave symmetry
which arises for $\gamma=0.40$ and then fixing (d)
$\lambda_{\mathrm{latt}}=0$, and (e) $\lambda_{\mathrm{latt}}=0.18$. (f)
Behavior of $\bar{\Delta}^{(d)}_{n}(\theta)$ for $\theta=0$ as a function of
$\lambda_{\mathrm{latt}}$ and $\omega_{n}$. Notice that the _s_ -wave and _d_
-wave SC gap display similar features as $\lambda_{\mathrm{latt}}$ approaches
$\lambda_{\mathrm{latt},c}$.

In order to obtain the effective field description of the nematic interaction
between the itinerant fermions, we consider the one-loop correction of the
nematic propagator by the particle-hole polarization bubble, in addition to
integrating out the acoustic phonons from the action corresponding to the
Hamiltonian $H_{\text{n-latt}}$. As a result, if we set the density of states
to its Fermi level value $N_{F}$, the bosonic propagator at the bare QCP
distance $r_{0}$ evaluates to
$D^{-1}=N^{-1}_{F}(r_{0}+Kq^{2}+\Pi_{\mathrm{elec}})+\Pi_{\mathrm{latt}}$,
where $\Pi_{\mathrm{elec}}$ and $\Pi_{\mathrm{latt}}$ refer, respectively, to
the particle-hole and lattice polarization bubbles. They are given by

$\displaystyle\Pi_{\mathrm{elec}}(\mathbf{q},\Omega_{m})$
$\displaystyle=(N_{F}g)^{2}\cos^{2}(2\varphi)\frac{|\Omega_{m}|}{v_{F}q},$ (5)
$\displaystyle\Pi_{\mathrm{latt}}(\mathbf{q},\Omega_{m})$
$\displaystyle=\frac{(\gamma_{2}+\gamma_{3})g^{2}_{\mathrm{latt}}}{\gamma^{2}_{1}-\Upsilon^{2}(\varphi)}\bigg{[}\cos^{2}(2\varphi)-\frac{\gamma_{1}+\gamma_{3}}{\gamma_{2}+\gamma_{3}}\bigg{]}.$
(6)

Furthermore, the parameters in the above equation turn out to be
$\gamma_{1}=C_{11}+C_{66}$, $\gamma_{2}=C_{11}-C_{66}$,
$\gamma_{3}=C_{12}+C_{66}$, and
$\Upsilon\left(\varphi\right)=\sqrt{\frac{1}{2}(\gamma_{2}^{2}+\gamma_{3}^{2})+\frac{1}{2}(\gamma_{2}^{2}-\gamma_{3}^{2})\cos(4\varphi)}$
de Carvalho and Fernandes (2019). In what follows, we consider systems in
which the lattice is equally strong to distortion fluctuations in both the
$B_{1g}$ and $B_{2g}$ channels. This implies that $(C_{11}-C_{12})/2=C_{66}$
and, consequently, leads to $\gamma_{2}=\gamma_{3}$. Therefore, the bosonic
nematic mass becomes angular dependent and given by
$r(\varphi)=(r_{0}-r_{0,c})/(Kk^{2}_{F})+\lambda_{\mathrm{latt}}\cos^{2}(2\varphi)$,
where $r_{0,c}\equiv\frac{N_{F}g_{\mathrm{latt}}^{2}}{C_{66}}$ and
$\lambda_{\mathrm{latt}}\equiv\frac{r_{0,c}}{2Kk^{2}_{F}}\left(1+\frac{C_{12}}{C_{11}}\right)$.
Since $r(\varphi)$ goes to zero only along the diagonals of the Brillouin
zone, the phenomenon of nematic quantum criticality becomes directional-
selective (see Ref. Paul and Garst (2017)).

_Self-consistent Eliashberg equations.–_ The emergence of the SC state in the
present model is studied in terms of the Migdal-Eliashberg theory Eliashberg
(1960); Scalapino  _et al._ (1966); Carbotte (1990). As expressed in Eqs.
(1)–(3), we consider that the attractive interaction between the conduction
electrons is provided solely by the collective fluctuations of the nematic
field $\varphi(\mathbf{q})$, although we note that other effects such as the
Coulomb repulsion and the electron-phonon interaction might also become
relevant in the limit in which lattice distortions are strong. In order to
derive the Eliashberg equations, we first express the fermionic operators
$\psi_{\sigma}(\mathbf{k})$ and $\psi^{\dagger}_{\sigma}(\mathbf{k})$ in terms
of the Nambu spinor
$\Psi(\mathbf{k})=(\psi_{\uparrow}(\mathbf{k}),\;\psi^{\dagger}_{\downarrow}(-\mathbf{k}))^{T}$.
As a result, the interacting fermionic propagator related to
$\Psi(\mathbf{k})$ becomes
$\bm{\mathcal{G}}^{-1}(\mathbf{k},\omega_{n})=i\omega_{n}\bm{\tau}^{0}-\xi_{\mathbf{k}}\bm{\tau}^{z}-\bm{\Sigma}(\mathbf{k},\omega_{n})$,
where $\bm{\tau}^{0}$ and $\bm{\tau}^{x,y,z}$ are, respectively, the identity
and Pauli matrices in the Nambu space, and
$\bm{\Sigma}(\mathbf{k},\omega_{n})$ refers to the self-energy of the
conduction electrons. Taking into account the effect of vertex interactions
with bosonic propagators renormalized by one-loop particle-hole and lattice-
distortion corrections, the fermionic self-energy evaluates to

$\displaystyle\bm{\Sigma}(\mathbf{k},\omega_{n})$
$\displaystyle=g^{2}T\sum_{\mathbf{k}^{\prime},n^{\prime}}f^{2}\bigg{(}\frac{\mathbf{k}+\mathbf{k}^{\prime}}{2}\bigg{)}D(\mathbf{k}-\mathbf{k}^{\prime},\omega_{n}-\omega_{n^{\prime}})$
$\displaystyle\times\bm{\tau}^{z}\bm{\mathcal{G}}(\mathbf{k}^{\prime},\omega_{n^{\prime}})\bm{\tau}^{z}.$
(7)

The above result is formally obtained by summing an infinite series involving
rainbow Feynman diagrams with non-interacting fermionic propagators.

The parametrization of the fermionic the self-energy in Eq. (Superconductivity
near a nematoelastic quantum critical point) according to
$\bm{\Sigma}(\mathbf{k},\omega_{n})=i\omega_{n}[1-Z_{n}(\mathbf{k})]\bm{\tau}^{0}+\chi_{n}(\mathbf{k})\bm{\tau}^{z}+\Phi_{n}(\mathbf{k})\bm{\tau}^{x}$
yields three coupled Eliashberg equations relating the mass renormalization
function $Z_{n}(\mathbf{k})$, the energy shift $\chi_{n}(\mathbf{k})$, and the
anomalous self-energy $\Phi_{n}(\mathbf{k})$ that encodes the information on
the pairing mechanism. To simplify our analysis, we restrict the momentum
dependence of $Z_{n}(\mathbf{k})$, $\chi_{n}(\mathbf{k})$, and
$\Phi_{n}(\mathbf{k})$ to the angle $\theta$ of the Fermi momentum
$\mathbf{k}_{\theta}=k_{F}(\cos\theta\hat{\mathbf{x}}+\sin\theta\hat{\mathbf{y}})$
and evaluate the momentum integrals perpendicular to the Fermi surface by
setting the density of states to $N_{F}$, according to the Migdal-Eliashberg
approximation. This assumption is reasonable because the nematic boson is
assumed to be a slow mode compared to a fermion, i.e., the boson dispersion is
much smaller than its fermionic counterpart. We also make the linearization of
these equations with respect to the anomalous self-energy $\Phi(\theta)$,
because this approximation is sufficient to determine the $T_{c}$ and the gap
dependence on $\theta$ at the SC transition. As a result, we obtain that the
energy shift becomes identically zero, i.e., $\chi_{n}(\theta)=0$. The
remaining linearized Eliashberg equations evaluate to

$\displaystyle Z_{n}(\theta)-1=$
$\displaystyle\frac{\pi\overline{g}^{2}}{2N_{F}}\frac{T}{\omega_{n}}\sum\limits_{\omega_{n^{\prime}}}\int^{2\pi}_{0}\frac{d\theta^{\prime}}{2\pi}D(\mathbf{k}_{\theta}-\mathbf{k}^{\prime}_{\theta^{\prime}},\omega_{n}-\omega_{n^{\prime}})$
$\displaystyle\times
f^{2}\bigg{(}\frac{\mathbf{k}_{\theta}+\mathbf{k}^{\prime}_{\theta^{\prime}}}{2}\bigg{)}\operatorname{sgn}(\omega_{n^{\prime}}),$
(8) $\displaystyle Z_{n}(\theta)\Delta_{n}(\theta)=$
$\displaystyle\frac{\pi\overline{g}^{2}}{2N_{F}}T\sum\limits_{\omega_{n^{\prime}}}\int^{2\pi}_{0}\frac{d\theta^{\prime}}{2\pi}D(\mathbf{k}_{\theta}-\mathbf{k}^{\prime}_{\theta^{\prime}},\omega_{n}-\omega_{n^{\prime}})$
$\displaystyle\times
f^{2}\bigg{(}\frac{\mathbf{k}_{\theta}+\mathbf{k}^{\prime}_{\theta^{\prime}}}{2}\bigg{)}\frac{\Delta_{n^{\prime}}(\theta^{\prime})}{|\omega_{n^{\prime}}|},$
(9)

where $\overline{g}\equiv N_{F}g$ and
$\Delta_{n}(\theta)\equiv\Phi_{n}(\theta)/Z_{n}(\theta)$ corresponds to the SC
gap. Therefore, by inserting Eq. (8) into (9), this reduces the pairing
problem to the solution of a 2D integral equation for $\Delta_{n}(\theta)$.

Figure 3: SC critical temperature $\overline{T}_{c}\equiv T_{c}/E_{F}$ as a
function of the effective electron interaction represented by $\gamma$, and
the nematoelastic coupling $\lambda_{\mathrm{latt}}$. Panels (a) and (b) show
the _s_ -wave and _d_ -wave behavior of $T_{c}$ for
$\lambda_{\mathrm{latt}}=0$ and $\lambda_{\mathrm{latt}}=0.15$, respectively.
Notice that for $\lambda_{\mathrm{latt}}>0$, the SC state appears only for
$\gamma>\gamma_{c}$, where the critical interaction $\gamma_{c}$ increases
with $\lambda_{\mathrm{latt}}$. Besides, the _s_ -wave $T_{c}$ is always
larger than its _d_ -wave counterpart and has a larger $\gamma_{c}$ for the
same value of the nematoelastic coupling. (c) In the
$\gamma\gg\lambda_{\mathrm{latt}}$ limit, the difference between the _s_ -wave
and _d_ -wave SC critical temperature $T_{c}$ approaches a constant value,
which is independent of $\lambda_{\mathrm{latt}}$.

We point out here that, instead of approximating the renormalization mass
function $Z_{n}(\theta)$ by its scaling form at zero temperature to solve the
gap equation (9), we will employ its exact expression at the transition
temperature $T_{c}$. However, this will require the numerical evaluation of
the angular integral in Eq. (8). On the other hand, the Matsubara sum on the
right-hand side of this equation can be put in closed-form. By evaluating it,
we obtain

$\displaystyle
Z_{n}(\theta)=1-\frac{1}{\overline{\omega}_{n}}\int_{\theta^{\prime}}\bigg{\\{}q(\theta,\theta^{\prime})\bigg{[}\psi_{0}\bigg{(}\frac{q(\theta,\theta^{\prime})\mathcal{C}(\theta,\theta^{\prime})}{2\pi\gamma\overline{T}\cos^{2}(\theta+\theta^{\prime})}+1\bigg{)}$
$\displaystyle-\psi_{0}\bigg{(}\frac{q(\theta,\theta^{\prime})\mathcal{C}(\theta,\theta^{\prime})}{2\pi\gamma\overline{T}\cos^{2}(\theta+\theta^{\prime})}+n+1\bigg{)}\bigg{]}-\frac{\pi\gamma\overline{T}\cos^{2}(\theta+\theta^{\prime})}{\mathcal{C}(\theta,\theta^{\prime})}\bigg{\\}},$
(10)

where $\psi_{0}(x)$ is the digamma function, $q(\theta,\theta^{\prime})\equiv
2|\sin[(\theta-\theta^{\prime})/2]|$,
$\mathcal{C}(\theta,\theta^{\prime})\equiv r+q^{2}(\theta,\theta^{\prime})$,
$\gamma=\overline{g}^{2}/(Kk^{2}_{F})$ is the effective electron coupling,
$E_{F}=v_{F}k_{F}$ is the Fermi energy, $\overline{T}=T/E_{F}$
($\overline{\omega}_{n}=\omega_{n}/E_{F}$) is the reduced temperature
(frequency), and
$\int_{\theta}(\cdots)\equiv\int^{2\pi}_{0}\frac{d\theta}{2\pi}(\cdots)$. Note
that to obtain the expression in Eq. (Superconductivity near a nematoelastic
quantum critical point), we made the substitution
$f^{2}[(\mathbf{k}_{\theta}+\mathbf{k}^{\prime}_{\theta^{\prime}})/2]=\cos^{2}(\theta+\theta^{\prime})$.
We should also point out that the distance to the nematic QCP $r=r(\varphi)$
depends on the scattering angles $\theta$ and $\theta^{\prime}$ through
$\varphi=(\theta+\theta^{\prime})/2-\pi/2$. Finally, the last term on the
right-hand side of Eq. (Superconductivity near a nematoelastic quantum
critical point) diverges as the QCP is approached. However, this divergent
term is required to cancel out another divergent contribution, which appears
in the Matsubara sum of Eq. (9) when $n^{\prime}=n$.

_Numerical results.–_ The critical temperature $T_{c}$ and the gap function
$\Delta_{n}(\theta)$ are obtained by solving numerically the integral equation
(9) by the Nyström method Press  _et al._ (1992). In this case, we transform
this equation into the eigenvalue problem
$\sum_{j,m}\mathcal{K}_{n,m}(\theta_{i},\theta_{j})\Delta_{m}(\theta_{j})=\varepsilon\Delta_{n}(\theta_{i})$
and demand that the highest eigenvalue $\varepsilon$ should be equal to zero.
All results presented here are obtained by utilizing a cutoff with $N=30$
Matsubara frequencies and then discretizing the interval $[0,2\pi[$ for the
variable $\theta$ into 1000 points. For all results appearing in this work, we
also consider that the system is located at the QCP emerging from lattice
effects, i.e., at $r_{0}=r_{0,c}$. Fluctuations of the latter are taken by
changing only the effective coupling $\lambda_{\mathrm{latt}}$. If we set the
microscopic interaction $g_{\mathrm{latt}}$ to a constant, this situation is
obtained by fixing the elastic constant $C_{66}$ and then varying either
$C_{11}$ or $C_{12}$.

Figure 2 describes the behavior of the SC gap for _s_ -wave and _d_ -wave
symmetries at the transition to the SC state, as a function of
$\lambda_{\mathrm{latt}}$. As the nematoelastic interaction is switched on,
the SC gap for both symmetries begins to decay along the whole Fermi surface.
This implies a monotonic behavior in frequency for the SC gap
$\Delta^{(s,d)}_{n}(\theta)$ at any value of momentum defined by $\theta$. At
the critical point $\lambda_{\mathrm{latt}}=\lambda_{\mathrm{latt},c}$, where
the SC state undergoes a transition to a FL (QCP∗), the gap
$\Delta^{(s,d)}_{n}(\theta)$ becomes degenerate in frequency.

Figure 4: Dependence of the SC critical temperature for _s_ -wave pairing on
the effective electron interaction $\gamma$, and the nematoelastic coupling
$\lambda_{\mathrm{latt}}$. (a) For a fixed value of $\gamma$, $T^{(s)}_{c}$
decreases monotonically as $\lambda_{\mathrm{latt}}$ increases, and becomes
zero at a critical coupling
$\lambda^{(s)}_{\mathrm{latt},c}\sim\gamma^{\eta_{s}}$. The yellow and blue
regions refer, respectively, to the emergence of FL and NFL behavior in the
absence of SC; they are separated by the crossover (solid) line, which is
given by $T_{\mathrm{FL}}\sim\lambda^{3/2}_{\mathrm{latt}}E_{F}$. (b) The
behavior of
$(\overline{T}^{(s)}_{c,0}-\overline{T}^{(s)}_{c})/\overline{T}^{(s)}_{c,0}$
as a function of
$(\lambda_{\mathrm{latt},c}-\lambda_{\mathrm{latt}})/\lambda_{\mathrm{latt},c}$
is characterized by a universal function, where
$\overline{T}^{(s)}_{c,0}=\overline{T}^{(s)}_{c}(\gamma)$ is the SC critical
temperature for _s_ -wave pairing in the absence of nematoelastic coupling.
The inset shows the power-law dependence of $\overline{T}^{(s)}_{c}$ on
$(\lambda_{\mathrm{latt},c}-\lambda_{\mathrm{latt}})$ for $\gamma=0.8$. The
same feature is also found in the transition region for other values of
$\gamma$.

In Fig. 3, we display the dependence of $T_{c}$ on the effective interaction
$\gamma$ for both _s_ -wave and _d_ -wave symmetries, when
$\lambda_{\mathrm{latt}}$ is held fixed. For $\lambda_{\mathrm{latt}}=0$, we
find in agreement with Ref. Klein and Chubukov (2018) that both SC critical
temperatures for these gap symmetries approach zero as a power-law given by
$T^{(s)}_{c}\sim T^{(d)}_{c}\propto\gamma^{2}E_{F}$ in the weak coupling
regime. Besides, $T^{(s)}_{c}$ is always larger than $T^{(d)}_{c}$ and this
behavior also occurs in the strong coupling limit, although in this case the
difference between $T^{(s)}_{c}$ and $T^{(d)}_{c}$ turns out to be more
appreciable [see Fig. 3(a)]. As $\lambda_{\mathrm{latt}}$ becomes finite, the
system evolves to a SC state only if the electron interaction $\gamma$ is
larger than a critical value $\gamma_{c}$, which according to the results in
Figs. 3(b) and 3(c) increases with $\lambda_{\mathrm{latt}}$. This bears some
resemblance with the $T_{c}$ behavior of an effective model for low-density
systems, when one considers the combined effect of electron-phonon attraction
and electron-electron repulsion Phan and Chubukov (2022); *Pimenov-
arXiv(2021). Similarly to what happens with the electrical resistivity of a 2D
system described by the present model de Carvalho and Fernandes (2019); Vieira
_et al._ (2020), the effect of $\lambda_{\mathrm{latt}}$ on the SC critical
temperature becomes irrelevant in the regime
$\gamma\gg\lambda_{\mathrm{latt}}$. In fact, this can be seen in Fig. 3(c),
which shows the tendency for the difference $T^{(s)}_{c}-T^{(d)}_{c}$ to
saturate at a constant, independent of $\lambda_{\mathrm{latt}}$, as $\gamma$
flows to the strong coupling regime.

We also study the behavior of $T_{c}$ for _s_ -wave and _d_ -wave pairing by
fixing $\gamma$ and then varying the nematoelastic coupling. As displayed in
Fig. 4(a), the SC critical temperature $T^{(s)}_{c}$ for _s_ -wave pairing
decreases monotonically with $\lambda_{\mathrm{latt}}$ and then becomes zero
for nematoelastic couplings above a certain critical value
$\lambda_{\mathrm{latt},c}$. Most importantly, we also find that $T^{(s)}_{c}$
depends on $(\lambda_{\mathrm{latt},c}-\lambda_{\mathrm{latt}})$ as a power-
law in the vicinity of $\lambda_{\mathrm{latt},c}$, as seen in the inset of
Fig. 4(b). We also verified that this dependence does not change, as $\gamma$
varies from the weak to the strong coupling regime. In fact, if we restrict
the nemato-elastic coupling to the interval
$0\leqslant\lambda_{\mathrm{latt}}\leqslant\lambda_{\mathrm{latt},c}$ [see
Fig. 4(b)], the numerical solution of the Eliashberg equation for
$\Delta_{n}(\theta)$ yields:

$T^{(\ell)}_{c}(\gamma,\lambda_{\mathrm{latt}})=T^{(\ell)}_{c,0}(\gamma)\bigg{[}1-\mathscr{F}_{\ell}\bigg{(}1-\frac{\lambda_{\mathrm{latt}}}{\lambda^{(\ell)}_{\mathrm{latt},c}(\gamma)}\bigg{)}\bigg{]},$
(11)

where $\ell\in\\{s,d\\}$, $T^{(\ell)}_{c,0}(\gamma)$ denotes the SC critical
temperature in the absence of nematoelastic coupling, and
$\mathscr{F}_{\ell}(x)$ refers to a universal scaling function with the
asymptotic dependence

$\mathscr{F}_{\ell}(x)=\begin{dcases}1-\mathscr{A}_{\ell}x^{\alpha_{\ell}},&\text{if
}x\rightarrow 0^{+},\\\ \mathscr{B}_{\ell}(1-x)^{\beta_{\ell}},&\text{if
}x\rightarrow 1^{-}.\end{dcases}$ (12)

Figure 4(b) shows the features of $\mathscr{F}_{\ell}(x)$ for _s_ -wave
pairing, when one varies its argument from zero to one. In addition, the
behavior of $\mathscr{F}_{\ell}(x)$ for _d_ -wave pairing is quite similar to
the one presented in Fig. 4(b), although the parameters of that function
differ for each gap symmetry. In fact, according to our numerical solution,
they evaluate to $\mathscr{A}_{s(d)}\approx 0.68959\;(0.71402)$,
$\mathscr{B}_{s(d)}\approx 1.24779\;(1.46962)$, $\alpha_{s(d)}\approx
1.42043\;(1.4689)$, and $\beta_{s(d)}\approx 0.79867\;(0.872001)$. Although
our calculations found that the _s_ -wave and _d_ -wave $T_{c}$ are
characterized by different exponents $\alpha_{\ell}$ and $\beta_{\ell}$, we
cannot rule out the possibility that they are degenerate. In fact, since
$\alpha_{s}$ and $\alpha_{d}$ (or $\beta_{s}$ and $\beta_{d}$) are to some
extent not far numerically from each other, this particular question can only
be completely settled by either working with numerical solutions containing
even larger values for the Matsubara cutoff or by finding an exact solution
for the Eliashberg equation for $\Delta_{n}(\theta)$, which are both beyond
the scope of the present work. Finally, we also found numerically that the
critical nematoelastic coupling has a power-law dependence on $\gamma$ given
by $\lambda^{(\ell)}_{\mathrm{latt},c}\sim\gamma^{\eta_{\ell}}$, with exponent
$\eta_{s(d)}\approx 1.78375\;(1.32959)$. Consequently, the _s_ -wave SC state,
when compared to the _d_ -wave state, is indeed less susceptible to lattice
distortions.

_Conclusions and outlook.–_ We have studied the emergence of spin-singlet SC
with either _s_ -wave or _d_ -wave symmetry in a 2D electronic system near a
nematic QCP, in which the anisotropic nematic fluctuations couple linearly to
the elastic modes of the lattice. We find that the main effect of this
interaction is to reduce $T_{c}$ and, for large enough nematoelastic
couplings, induce a SC-FL quantum phase transition. This transition, along
with the SC-NFL transition that takes place for small values of
$\lambda_{\mathrm{latt}}$, are described by power-laws characterized by
distinct critical exponents, which may also differ for _s_ -wave and _d_ -wave
pairings. Lattice effects were shown to play a crucial role on the phase
diagram of systems with only a structural QPT to an orthorhombic state Cowley
(1976); Qi and Xu (2009); Fernandes  _et al._ (2010); Cano  _et al._ (2010);
Labat and Paul (2017); Karahasanovic and Schmalian (2016), as is the case of
the chemically substituted iron-chalcogenide FeSe1-xSx. Therefore, this
compound represents to date the best physical platform to observe the features
investigated here. Indeed, it is observed experimentally in the phase diagram
of FeSe0.89S0.11 that $T_{c}$ suffers a depletion at the onset of nematic
order Reiss  _et al._ (2020). According to our present results, this might be
related to an increase of the nematoelastic coupling as the nematic QCP is
approached. This conclusion is also supported by several recent experimental
Reiss  _et al._ (2020); Bristow  _et al._ (2020); Chibani  _et al._ (2021) and
theoretical works de Carvalho and Fernandes (2019); Wang and Berg (2019);
Vieira  _et al._ (2020) on the behavior of the electrical resistivity
exhibited by this correlated material.

_Acknowledgments.–_ H.F. acknowledges funding from CNPq under grant numbers
310710/2018-9 and 311428/2021-5.

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