# Exact Black Hole Solutions with a Conformally Coupled Scalar Field and
Dynamic Ricci Curvature in $f(R)$ Gravity Theories

Thanasis Karakasis<EMAIL_ADDRESS>Physics Division, National Technical
University of Athens, 15780 Zografou Campus, Athens, Greece.  Eleftherios
Papantonopoulos<EMAIL_ADDRESS>Physics Division, National Technical
University of Athens, 15780 Zografou Campus, Athens, Greece.  Zi-Yu Tang
<EMAIL_ADDRESS>School of Fundamental Physics and Mathematical Sciences,
Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Bin Wang<EMAIL_ADDRESS>School of Aeronautics and Astronautics,
Shanghai Jiao Tong University, Shanghai 200240, China  Center for Gravitation
and Cosmology, College of Physical Science and Technology, Yangzhou
University, Yangzhou 225009, China

###### Abstract

We report exact black hole solutions in asymptotically flat or (A)dS four-
dimensional spacetime with a conformally coupled self-interacting scalar field
in $f(R)$ gravity. We first consider the asymptotically flat model
$f(R)=R-2\alpha\sqrt{R}$ and derive an exact black hole solution. Then, we
consider the asymptotically (A)dS model
$f(R)=R-2\Lambda-2\alpha\sqrt{R-4\Lambda}$ and derive an exact black hole
solution. In both cases the modified gravity parameter $\alpha$, which has the
dimension of the inverse mass, cannot be set to zero and the self-interacting
potential is determined from the Klein-Gordon equation, preserving the
conformal invariance. The thermodynamics of the solutions is also studied.

###### Contents

  1. I Introduction
  2. II Black Hole Solutions in Conformal Gravity Theories
  3. III Black Hole Solutions in Conformal $f(R)$ Gravity Theories
    1. III.1 $f(R)=R-2\alpha\sqrt{R}$
      1. III.1.1 Neutral Solution
      2. III.1.2 Solution with Charge
    2. III.2 $f(R)=R-2\Lambda-2\alpha\sqrt{R-4\Lambda}$
      1. III.2.1 Neutral Solution
      2. III.2.2 Solution with Charge
  4. IV Conclusions

##  I Introduction

The first exact black hole solution with a scalar field as a matter field was
found by Bocharova, Bronnikov and Melnikov and independently by Bekenstein,
called BBMB black hole [1]. The scalar field is conformally coupled to
gravity, resulting in the vanishing of the trace of the energy-momentum
tensor, which means that the scalar curvature is constant and in this
particular case in the absence of cosmological constant is zero. The resulting
spacetime is the external Reissner-Nordström (RN) spacetime and the scalar
field diverges at the black hole horizon. It was also shown in [2] that this
solution is unstable under scalar perturbations. Later, a scale was introduced
to the theory via a cosmological constant in [3] and also a quartic scalar
potential that does not break the conformal invariance of the action, which
gives a very simple relation between the scalar curvature and the cosmological
constant. In this case, the scalar field does not diverge at the horizon, but
the solution is found to be unstable [4].

Regarding the minimal coupling case, the first exact black hole solution was
presented in [5], the MTZ black hole. The scalar potential is fixed ad hoc,
the geometry of the solution is hyperbolic and the scalar field remains finite
at the black hole horizons. In [6] the electrically charged case was
discussed. In [7], a potential that breaks the conformal invariance of the
action of the MTZ black hole in the Jordan frame was considered and new black
hole solutions where derived. In [8] the scalar field was fixed ad hoc and
novel black hole solutions were investigated, letting the scalar potential to
be determined from the equations and in [9] the electrically charged case is
considered. In [10], asymptotically (anti) de Sitter black holes and wormholes
with a self-interacting scalar field in four dimensions were investigated. In
[11, 12, 13, 14, 15] black holes with non-minimal derivative coupling were
studied. However, the scalar field which was coupled to Einstein tensor should
be considered as a particle living outside the horizon of the black hole
because it blows up on the event horizon. In [16] super-entropic black holes
with Immirzi hair were investigated and Plebanski-Demianski solutions in
quadratic gravity with conformally coupled scalar fields were also found [17].
Black holes in Lanczos-Lovelock gravity theories with a non-minimally coupled
scalar field were recently discussed [18] and hairy black holes in disformal
gravity theories were investigated in [19].

In the context of $f(R)$ gravity (for a review see [21, 20]) several black
hole solutions have been found recently. Vacuum solutions were discussed in
[22, 23, 24, 25, 26], while charged black hole solutions were also found in
[35, 27, 28, 29, 31, 32, 33, 34]. In [36], dynamical black holes,
characterized by time-varying apparent horizons, in various theories of
gravity, including the $f(R)$ theory were investigated. The possibility of
scalarization of black holes in the context of $f(R)$ gravity was discussed in
[37] and $(2+1)$-dimensional black holes with a minimally coupled self-
interacting scalar field in the context of $f(R)$ gravity was discussed in
[38]. Recently non-trivial black hole solutions were investigated [39]. These
solutions are considered non-trivial in the sense that $g_{tt}g_{rr}\neq-1$
and the integration of the field equations is achieved by fixing the
$f_{R}(r)$ function. Finally in [40] the uniqueness of non-trivial spherically
symmetric black hole solution in special classes of $f(R)$ gravitational
theory was discussed.

The viability of any theory of gravity should also be tested against the
astrophysical observations. Differences between general relativity (GR) and
alternative theories are expected to occur for strong gravitational fields,
such as the ones created by different compact objects like neutron stars,
strange stars and black holes. In [41] an $f(R)$ modified theory of the form
$f(R)=R+\alpha R^{2}$ was investigated. Studying neutron stars in which strong
gravity effects are non-negligible, it was found that the neutron stars within
these theories can differ significantly from their GR counterpart which makes
them a very good candidate to test $f(R)$ theories on astrophysical scales
[42].

In this $f(R)$ modified theory studying non-perturbatively and self-
consistently the structure of neutron stars, bounds on the $\alpha$ parameter
were imposed [43]. Employing various equations of state for the neutron star,
the mass-radius relations were obtained constraining the $\alpha$ parameter
resulting at large deviations from GR. Then there was a discussion of the
choice of equation of state [44, 45, 46] trying to impose a realistic equation
of state to search for predictions or derive relations that are independent of
the equation of state.

In our previous work [38] we considered $(2+1)$-dimensional $f(R)$ gravity
with a self interacting scalar field as a matter field. The motivation for
considering an explicit scalar field was that in $f(R)$ gravity theories if a
conformal transformation is applied from the Jordan frame to the Einstein
frame then, a new scalar field with a potential appears in the Lagrangian. The
generated scalar-tensor theory has a geometric (gravitational) scalar field
which however cannot dress a $f(R)$ black hole with hair [48, 47, 49]. Without
specifying the form of the $f(R)$ function we derived the field equations and
we showed that the $f(R)$ model has a direct contribution from the scalar
field.

At first we considered the case, where
$f_{R}(r)=1-\int\int\phi^{\prime}(r)^{2}drdr$. Integrating with respect to the
Ricci scalar we obtained a pure Einstein-Hilbert term and another term that
depended on the scalar field. The scalar curvature was dynamical and due to
its complexity it was difficult to obtain an exact form of the $f(R)$
function. Using asymptotic approximations, we showed that the scalar charges
make the theory to deviate form Einstein’s Gravity. Then an exact black hole
solution dressed with a scalar hair was found, in which the scalar charge
appears in the $f(R)$ function and the thermodynamics of the solution was
studied. We further considered a pure $f(R)$ theory supported by the scalar
field. We showed that the thermodynamic and observational constraints require
that the pure $f(R)$ theory to be builded with a phantom scalar field. The
scalar charge is the one that determines the behaviours of the solution.

In this work we will extent our previous work to a more realistic model. We
will work in $(3+1)$-dimensions with the aim to generate a compact object with
regular matter, i.e. a black hole dressed with scalar hair. The common
practice in constructing such objects is to choose a particular equation of
state, expressing the matter content of the $f(R)$ theory. Instead of that, we
will introduce matter as a conformably coupled scalar field and a dynamic
scalar curvature. This particular type of theory, $f(R)$ gravity and non-
minimally coupled scalar fields as matter in $(3+1)$-dimensions, has been
previously considered for cosmological purposes [50, 51]. In our study, the
scalar field is introduced in the action as a matter field in the way it was
done in the GR context [1, 3] and we study its effect on a metric ansatz
solving the field equations. In [48] there is reported a black hole solution
with a conformally coupled scalar field for $f(R)=R^{2}$, but since $R^{2}$
describes an exact (A)dS space [58], the solution turns out to be identical to
the one reported in [3].

At first we will discuss the well known GR cases of the BBMB black hole [1]
and the de Sitter black hole with a conformally coupled scalar field [3], in
order to compare our modified gravity solutions with the Einstein Gravity
ones. In the BBMB black hole model, the action, except for the Ricci scalar
$R$ includes the kinetic energy of the scalar field and its conformal coupling
to $R$. In our model a general $f(R)$ function is introduced instead of the
Ricci scalar.

In the literature there are various choices for the $f(R)$ function (for an
extensive review of the $f(R)$ gravity theories see [52, 53].) These choices
depend on the requirements of having consistent gravity theories, free of
ghosts and tachyonic instabilities. The parameters introduced in the $f(R)$
functions are constrained by the cosmological observations. In our study we
discuss the profiles $f(R)=R-2\alpha\sqrt{R}$ and
$f(R)=R-2\Lambda-2\alpha\sqrt{R-4\Lambda}$. These particular profiles allow us
to find exact black hole solutions dressed with a scalar field. The solutions
respect the conformal invariance and the self-interacting potential of the
scalar field is determined from the field equations, respecting the conformal
invariance. We checked the trace of the resultant energy-momentum tensor and
find it vanishing, as a result of the conformal invariance. The scalar field
diverges between the inner and outer horizon which is what happens in the
context of GR [3]. We also studied the thermodynamics of the solutions and we
found that an electric charge has to be introduced, for the solution to have
positive entropy.

As it happens in the astrophysical applications of the $f(R)$ gravity
theories, since we are in a strong gravity regime, the parameter $\alpha$
having dimensions of inverse mass, plays a decisive role in the construction
of compact objects like black holes. Solving the full system of field
equations we find that this parameter contributes to the mass of the black
hole, specifies the strength of the scalar field and influences the
thermodynamical properties of the solutions.

If we switch off the scalar field, these profiles of the $f(R)$ function can
generate analytic spherically symmetric black hole solutions that have been
extensively studied [22, 23, 30], while in [31] an electromagnetic field was
added. It is interesting to note that these black holes are characterized by
the parameter $\alpha$ that makes solutions deviate from the standard
solutions of GR. The Kretschmann scalar and squared Ricci tensor are shown to
depend on the parameter $\alpha$ which is not allowed to be zero. Also these
black hole solutions have interesting thermodynamical properties [31].

This paper is organized as follows: In section II we briefly discuss the black
hole solutions with a conformally coupled scalar field in the context of GR.
In section III we derive novel black hole solutions with a conformally coupled
scalar field and dynamic Ricci curvature. The thermodynamics of the solutions
is also studied. Finally in section IV we conclude.

##  II Black Hole Solutions in Conformal Gravity Theories

In this Section we discuss the GR black hole solutions with a scalar field
conformally coupled to gravity. The first solution was found by Bocharova,
Bronnikov and Melnikov and independently by Bekenstein, called the BBMB black
hole [1].

Consider the action in which a scalar field is conformally coupled to gravity

$S=\int d^{4}x\sqrt{-g}\left[\cfrac{1}{16\pi
G}R-\cfrac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi-\cfrac{1}{12}R\phi^{2}\right]~{}.$
(1)

By variation of the above action we obtain the Einstein equation and Klein-
Gordon equation respectively

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle 8\pi
GT_{\mu\nu}~{},$ (2) $\displaystyle\Box\phi$ $\displaystyle=$
$\displaystyle\cfrac{1}{6}R\phi~{},$ (3)

where the energy-momentum tensor is,

$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\cfrac{1}{2}g_{\mu\nu}\nabla^{\alpha}\phi\nabla_{\alpha}\phi+\cfrac{1}{6}\big{(}g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}+G_{\mu\nu}\big{)}\phi^{2}.$
(4)

In the following we will set $8\pi G=1$. The matter part of the action is
invariant under conformal transformations

$g_{\mu\nu}\to\Omega^{2}(x)g_{\mu\nu}~{},\hskip
28.45274pt\phi\rightarrow\Omega^{-1}(x)\phi~{}.$ (5)

As a result, the energy-momentum tensor is traceless and from the Einstein
equations we get $R=0$ for the scalar curvature.

Considering the spherically symmetric metric ansatz

$ds^{2}=-b(r)dt^{2}+b^{-1}(r)dr^{2}+r^{2}d\Omega^{2}~{},$ (6)

the $R=0$ relation gives

$r^{2}b^{\prime\prime}(r)+4rb^{\prime}(r)+2b(r)-2=0~{},$ (7)

from which we get

$b(r)=1+\cfrac{c_{1}}{r}+\cfrac{c_{2}}{r^{2}}~{},$ (8)

where $c_{1},c_{2}$ are constants of integration. Now, the Klein-Gordon
equation becomes

$\Box\phi=0~{}.$ (9)

From the Einstein equations we can obtain a relation for the scalar field

$2\phi^{\prime}(r)^{2}-\phi^{\prime\prime}(r)\phi(r)=0~{},$ (10)

which gives

$\phi(r)=\cfrac{1}{c_{3}r+c_{4}}~{},$ (11)

where $c_{3},c_{4}$ are constants of integration. Substituting the expressions
for the metric function and the scalar field to Einstein and Klein-Gordon
equations we obtain the BBMB black hole

$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\left(1-\cfrac{m}{r}\right)^{2}=1-\cfrac{2m}{r}+\cfrac{m^{2}}{r^{2}}~{},$
(12) $\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\pm\sqrt{6}\cfrac{m}{r-m}~{},$ (13)

where, $m$ is the mass of the black hole. To summarize, the metric function
has an extra $m^{2}/r^{2}$ term because of the presence of the scalar field
which resembles the extermal RN case, where the scalar field diverges at the
black hole horizon while the Kretschmann scalar is divergent at the origin
indicating a physical singularity. Also note that in some sense the scalar
field provides the mass of the black hole.

The thermodynamical properties of the BBMB black hole are interesting. The
Hawking temperature is zero, since it is given by

$T(r_{h})=\cfrac{b^{\prime}(r_{h})}{4\pi}=\frac{m}{2\pi
r^{2}}\left(1-\frac{m}{r}\right)\Big{|}_{r=m}=0~{},$ (14)

while the entropy [54, 55] becomes infinite

$S=\pi r_{h}^{2}\left(1-\cfrac{1}{6}\phi(r_{h})^{2}\right)~{},$ (15)

since the scalar field diverges at the event horizon. Therefore, the BBMB
black hole does not have the conventional thermodynamic properties of black
holes as it was discussed in [56] and since black holes should have non-zero
surface gravity (temperature) one may argue that the BBMB black hole is not a
black hole.

Hence, the BBMB black hole solution is not a hairy black hole because the
conformal symmetry does not allow the formation of finite hair at the horizon
with a scalar charge independent of the black hole mass that can be detected
asymptotically.

A generalization of the BBMB black hole solution was presented in [3].
Consider the action

$S=\int d^{4}x\sqrt{-g}\Bigg{(}\cfrac{R-2\Lambda}{16\pi
G}-\cfrac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\cfrac{1}{12}R\phi^{2}-\alpha\phi^{4}\Bigg{)}~{}.$
(16)

The resulting field equations are

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}-8\pi GT_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0~{},$ (17)
$\displaystyle\Box\phi-\cfrac{1}{6}R\phi-4\alpha\phi^{3}$ $\displaystyle=$
$\displaystyle 0~{},$ (18)

where

$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\cfrac{1}{2}g_{\mu\nu}\nabla^{\alpha}\phi\nabla_{\alpha}\phi+\cfrac{1}{6}\big{(}g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}+G_{\mu\nu}\big{)}\phi^{2}-\alpha
g_{\mu\nu}\phi^{4}~{}.$ (19)

The matter part of the action is invariant under the conformal transformations
(5). As a result, the energy-momentum tensor is traceless and in the presence
of the cosmological constant the scalar curvature is

$R=4\Lambda~{}.$ (20)

With the metric (6) the constant curvature relation (20) gives the metric
function

$b(r)=-\frac{\Lambda r^{2}}{3}+\frac{c_{2}}{r^{2}}+\frac{c_{1}}{r}+1~{}.$ (21)

From Einstein equations, we can obtain equation (10) for the profile of the
scalar field. Plugging the metric (21) and the scalar (11) back to Einstein
equations, the integration constants take particular values and the solution
becomes

$\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{3}{\pi}}\frac{\sqrt{G}M}{2GM-2r}~{},$ (22)
$\displaystyle b(r)$ $\displaystyle=$ $\displaystyle 1-\frac{\Lambda
r^{2}}{3}-\frac{2GM}{r}+\frac{G^{2}M^{2}}{r^{2}}=-\frac{\Lambda
r^{2}}{3}+\left(1-\cfrac{GM}{r}\right)^{2}~{}.$ (23)

In order to respect the conformal invariance, the parameter $\alpha$ is
specified, so the solution exists only for $\alpha=-2\pi G\Lambda/9$. There
are three horizons the inner, event and cosmological horizon and all possible
divergencies of the curvature invariants, the metric function and the scalar
field are hidden behind the event horizon. We note that we cannot have a black
hole solution for an AdS spacetime, since equation (23) is always positive.

The thermodynamics of this solution have been discussed in [54]. The
temperature of the black hole is given by

$T=\cfrac{1}{2\pi l}\sqrt{1-\cfrac{4M}{l}}~{},$ (24)

where $l$ is the dS radius $\Lambda=3/l^{2}$. The entropy at the black hole
horizon is negative, while the entropy at the cosmological horizon is positive
and the total entropy of the black hole is zero, since the two entropies have
the same absolute value.

The black hole solution discussed in [3] is a generalization of the BBMB black
hole solution in the presence of a positive cosmological constant. This
modification allows the scalar field to be finite on the event horizon
dressing the black hole with secondary scalar hair [57], but still the
thermodynamic properties of the solution indicate that the produced compact
object does not have a conventional thermodynamic behavior. In an attempt to
understand better the thermodynamical properties of the solution a charge was
introduced to the theory [3, 54].

##  III Black Hole Solutions in Conformal $f(R)$ Gravity Theories

As we have seen in the previous section in conformal gravity theories the
Ricci scalar plays an important role to obtain exact black holes. If it is
zero then no black hole with finite scalar hair can be produced, since the
scalar field is divergent at the event horizon. If it is proportional to a
positive cosmological constant then an exact black hole is generated with a
scalar field regular on the event horizon. However, in both cases we do not
get a black hole with the conventional thermodynamic properties. In this
section we will investigate the case of introducing a non-linear curvature
correction term to the Ricci scalar in the context of the $f(R)$ gravity
theory and study the thermodynamic properties of the resulting black hole
solutions.

We consider the action

$S=\int
d^{4}x\sqrt{-g}\left(\cfrac{f(R)}{2}-\cfrac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\cfrac{1}{12}R\phi^{2}-V(\phi)\right)~{},$
(25)

which consists of an arbitrarily differentiable function of the Ricci scalar
and a self-interacting, non-minimally coupled to gravity scalar field. The
factor of the non-minimal coupling is the conformal coupling factor and the
potential is arbitrary. We will determine the potential from the field
equations. The field equations are

$\displaystyle f_{R}R_{\mu\nu}-\cfrac{1}{2}f(R)g_{\mu\nu}+g_{\mu\nu}\Box
f_{R}-\nabla_{\mu}\nabla_{\nu}f_{R}$ $\displaystyle=$ $\displaystyle
T_{\mu\nu}~{},$ (26)
$\displaystyle\Box\phi-\cfrac{1}{6}R\phi-V^{\prime}(\phi)$ $\displaystyle=$
$\displaystyle 0~{},$ (27)

where $f_{R}=\cfrac{df(R)}{dR}$. The energy-momentum tensor is given by

$T_{\mu\nu}^{\phi}=\partial_{\mu}\phi\partial_{\nu}\phi-\cfrac{1}{2}g_{\mu\nu}\partial^{\alpha}\phi\partial_{\alpha}\phi+\cfrac{1}{6}\Big{(}g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}+G_{\mu\nu}\Big{)}\phi^{2}-g_{\mu\nu}V(\phi)~{}.$
(28)

Considering the same metric ansatz as in the GR cases (6), the components of
the Einstein field equation are

$0=r\left(r\left(b^{\prime}(\left(\phi\phi^{\prime}-3f^{\prime}_{R}\right)+3f_{R}b^{\prime\prime}-b\left(6f^{\prime\prime}_{R}+\phi^{\prime
2}-2\phi\phi^{\prime\prime}\right)+3f-6V\right)+b^{\prime}\left(6f_{R}+\phi^{2}\right)-12bf^{\prime}_{R}+4b\phi\phi^{\prime}\right)(b-1)\phi^{2}~{},$
(29)
$0=r\left(r\left(b^{\prime}\left(\phi\phi^{\prime}-3f^{\prime}_{R}\right)+3f_{R}b^{\prime\prime}+3b\phi^{\prime
2}+3f-6V\right)+b^{\prime}\left(6f_{R}+\phi^{2}\right)-12bf^{\prime}_{R}+4b\phi\phi^{\prime}\right)+(b-1)\phi^{2}~{},$
(30)
$0=r\left(-2\left(6rb^{\prime}f^{\prime}_{R}+b\left(6f^{\prime}_{R}+6rf^{\prime\prime}_{R}+r\phi^{\prime
2}\right)\right)+4\phi\left(\left(rb^{\prime}+b\right)\phi^{\prime}+rb\phi^{\prime\prime}\right)+\phi^{2}\left(2b^{\prime}+rb^{\prime\prime}\right)\right)+12f_{R}\left(rb^{\prime}+b-1\right)+6r^{2}(f-2V)~{}.$
(31)

The Klein-Gordon equation (27) for the metric (6) reads

$\frac{\phi(r)\left(r^{2}b^{\prime\prime}(r)+4rb^{\prime}(r)+2b(r)-2\right)}{6r^{2}}+b^{\prime}(r)\phi^{\prime}(r)+\frac{2b(r)\phi^{\prime}(r)}{r}+b(r)\phi^{\prime\prime}(r)-\frac{V^{\prime}(r)}{\phi^{\prime}(r)}=0~{}.$
(32)

The trace of Einstein equation (26) in tensor and differential form reads

$Rf_{R}-2f(R)+3\square f_{R}=\phi\Box\phi-R\phi^{2}/6-4V(\phi)~{},$ (33)
$0=r\left(2b^{\prime}\left(-9rf^{\prime}_{R}+3r\phi\phi^{\prime}+2\phi^{2}\right)+r\phi^{2}b^{\prime\prime}\right)+6f_{R}\left(r^{2}b^{\prime\prime}+4rb^{\prime}+2b-2\right)+2b\left(-9r\left(2f^{\prime}_{R}+rf^{\prime\prime}_{R}\right)+3r\phi\left(2\phi^{\prime}+r\phi^{\prime\prime}\right)+\phi^{2}\right)\\\
+12r^{2}(f-2V)-2\phi^{2}~{},$ (34)

while the trace of the energy-momentum tensor is

$T^{\mu}_{\mu}=\frac{\phi(r)\left(\phi(r)\left(r^{2}b^{\prime\prime}(r)+4rb^{\prime}(r)+2b(r)-2\right)+6r\left(\left(rb^{\prime}(r)+2b(r)\right)\phi^{\prime}(r)+rb(r)\phi^{\prime\prime}(r)\right)\right)}{6r^{2}}-4V(r)~{}.$
(35)

We will determine if the resulting solution is conformally coupled to gravity
by examining if the trace (35) vanishes.

In the following we will fix the $f(R)$ function, determine the resulting
black hole solutions and study their thermodynamic properties.

###  III.1 $f(R)=R-2\alpha\sqrt{R}$

####  III.1.1 Neutral Solution

We first consider the $f(R)$ model

$f(R)=R-2\alpha\sqrt{R}~{},$ (36)

where a non-linear curvature correction is added to the Einstein-Hilbert term
through the model parameter $\alpha$ which has the dimensions of $[L]^{-1}$
(inverse mass). Solving equations (29), (30), (31) and (32) we obtain the
following configurations

$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{3\alpha r}+\frac{3}{64\alpha^{2}r^{2}}~{},$
(37) $\displaystyle f_{R}(r)$ $\displaystyle=$ $\displaystyle 1-\alpha r~{},$
(38) $\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\frac{1-2\alpha
r}{r^{2}}~{},$ (39) $\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{6}}{1-4\alpha r}~{},$ (40) $\displaystyle V(r)$
$\displaystyle=$ $\displaystyle-\frac{4\alpha^{2}}{(4\alpha r-1)^{4}}~{},$
(41) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle-\frac{1}{9}\alpha^{2}\phi^{4}~{},$ (42) $\displaystyle R(r)$
$\displaystyle=$ $\displaystyle\cfrac{1}{r^{2}}~{}.$ (43)

The trace of the resulting energy-momentum tensor is zero, meaning that the
scalar field is conformally coupled to gravity and the scalar potential which
is obtained from the Klein-Gordon equation preserves the conformal invariance.
The metric function has two roots

$r_{\pm}=\frac{8\pm\sqrt{10}}{24\alpha}~{},$ (44)

which are both positive for $\alpha>0$, $r_{-}$ being the inner while $r_{+}$
being the outer horizon. There exists a simple pole (divergence) in the scalar
field function that lies between the horizons

$r_{-}<r_{divergence}<r_{+}~{}.$ (45)

The scalar field is finite at the event horizon and takes the value

$\phi(r_{+})=-\frac{6\sqrt{6}}{\sqrt{10}+2}~{}.$ (46)

The behavior of the scalar field is similar with the GR case [3]. The model is
stable only for $\alpha>0$ since then we have
$f_{RR}=\cfrac{\alpha}{2R^{3/2}}>0$ avoiding tachyonic instabilities [20]
which means that for $\alpha>0$ we can interpret this solution as a black hole
solution with a conformally coupled scalar field as matter in a viable $f(R)$
model. Therefore we will impose the condition $\alpha>0$ throughout the paper.

If the parameter $\alpha$ is non-zero then the Ricci curvature $R$ receives a
non-linear correction term and the field equations have the solution (37)-(43)
in which the only free parameter is $\alpha$. If $\alpha$ is zero then the
$f(R)$ is given by the Ricci curvature $R$ and then the field equations, as we
discussed in the previous section, give the BBMB solution (12), (13). In this
solution the mass is an integration constant which is fixed by the
gravitational potential and a mass-squared term appears because of the
presence of the scalar field. In the case of the $f(R)$ theory in which
gravity is stronger we have a similar behaviour. However, the mass of the
black hole is proportional to $1/\alpha$ and also this parameter gives an
effective charge term to the scalar field. A similar behaviour is found in the
vacuum and charged black hole solutions [22, 27, 31]. We note here that in
modified theories of gravity the strong-field gravity effects can potentially
introduce much greater differences in the parameters of local solutions in
these theories. These theories can pass the present weak-field gravitational
tests and exhibit non-perturbative strong-field deviations away from GR in
systems involving various compact objects [60, 61].

Figure 1: Quantities related to the $f(R)$ model: $f_{R}(r)$ and $f(r)$ for
different values of the parameter $\alpha$.

Figure 2: The metric function $b(r)$ (left) and the scalar field $\phi(r)$
(right) for different values of the parameter $\alpha$. Below, we plot the
scalar field while the vertical lines represent the positions of the horizons
for a particular value of $\alpha$.

Figure 3: The norm of the Weyl tensor (left) and the Kretschmann scalar
(right) as functions of $r$ for different values of the parameter $\alpha$.
Figure 4: The event (outer) horizon $r_{+}$ as a function of the parameter
$\alpha$.

In FIG. 1 we plot the $f_{R}(r)$ and $f(r)$ (the gravitational model as a
function of the radial coordinate). From $f(r)$ we can see that the
gravitational effects are stronger near the origin, since the Ricci scalar
diverges there, while tend rapidly to zero at large distances. In FIG. 2 we
plot the metric function and the scalar field in order to see the behaviour of
these functions in the regime (45). The metric function diverges at the origin
while tends rapidly to $1/2$ at large distances. The scalar field diverges
only between the inner and outer horizons and vanishes at large distances. We
should note that the scalar field does not dress the black hole with some kind
of hair. The only free parameter of the system is $\alpha$. The scalar field
does not have a scalar charge because of the conformal invariance but, it
acquires a charge from the curvature sector of the action. We compute the
Kretschmann scalar for the metric (37) and the norm of the Weyl tensor.

$\displaystyle R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}$
$\displaystyle=$
$\displaystyle\frac{55}{48\alpha^{2}r^{6}}-\frac{3}{4\alpha^{3}r^{7}}+\frac{63}{512\alpha^{4}r^{8}}+\frac{4}{3\alpha
r^{5}}+\frac{1}{r^{4}}~{},$ (47) $\displaystyle
C^{\alpha\beta\gamma\delta}C_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\frac{(1-4\alpha r)^{2}(4\alpha
r+9)^{2}}{768\alpha^{4}r^{8}}~{}.$ (48)

Both scalars diverge at the origin meaning that $r\rightarrow 0$ is a physical
singularity. The norm of the Weyl tensor also vanishes at the point where the
scalar field diverges. We present plots for the curvature invariants in FIG.
3. We also plot the event (outer) horizon of the black hole as a function of
the modified gravity parameter $\alpha$ in FIG. 4, where we can see that for
bigger values of $\alpha$, the black hole is formed closer to the origin of
the coordinates.

We have seen that the black hole solutions in GR with conformally coupled
scalar field do not have a conventional thermodynamical behavior, due to the
existence (and therefore modification of the area law for the entropy) of the
non-minimally coupled scalar field. We will discuss the thermodynamics of the
asymptotically flat black hole solution in the $f(R)$ gravity theory with the
choice $f(R)=R-2\alpha\sqrt{R}$. To compute the Hawking temperature and the
Bekenstein-Hawking entropy we use the following relations [59, 31, 54]

$\displaystyle T(r_{+})$ $\displaystyle=$
$\displaystyle\cfrac{b^{\prime}(r_{+})}{4\pi}~{},$ (49) $\displaystyle
S(r_{+})$ $\displaystyle=$
$\displaystyle\cfrac{\mathfrak{A}f_{R}(r_{+})}{4}=\cfrac{\mathfrak{A}}{4}\Big{(}f_{R}^{(\text{gravity})}(r_{+})+f_{R}^{(\text{matter})}(r_{+})\Big{)}=\cfrac{\mathfrak{A}}{4}\left(1-\alpha
r_{+}-\cfrac{1}{6}\phi(r_{+})^{2}\right)~{}.$ (50)

which are the Hawking temperature and the Bekenstein-Hawking entropy, where
$\mathfrak{A}$ denotes the area of the black hole $\mathfrak{A}=4\pi
r_{+}^{2}$. It is of major importance to note the fact that the entropy
acquires a multiplicative factor due to the existence of the non-minimal
coupling between Ricci scalar and scalar field. So, we do not only have the
modified gravity part but we also have contribution from the matter part of
the action, that modifies the area law for the entropy. The concrete
expressions of them are

$\displaystyle T(r_{+})$ $\displaystyle=$ $\displaystyle\frac{32\alpha
r_{+}-9}{384\pi\alpha^{2}r_{+}^{3}}=\frac{\left(37\sqrt{10}-80\right)\alpha}{243\pi}~{},$
(51) $\displaystyle S(r_{+})$ $\displaystyle=$ $\displaystyle\pi
r_{+}^{2}\left(1-\alpha r_{+}-\frac{1}{\left(1-4\alpha
r_{+}\right){}^{2}}\right)=\frac{5\left(191\sqrt{10}-848\right)\pi}{6912\alpha^{2}}~{}.$
(52)

It is clear that we cannot set the parameter $\alpha$ to zero. The Hawking
temperature is always positive and proportional to the model parameter
$\alpha$ while the Bekenstein-Hawking entropy is always negative and inversely
proportional to $\alpha^{2}$. Their figures are present in FIG. 5. The
thermodynamics of this solution if we set $\alpha\to 0$ behaves similar to the
BBMB black hole which we previously discussed.

Figure 5: The Hawking temperature $T(r_{+})$ (left) and the Bekenstein-Hawking
entropy $S(r_{+})$ (right) at the event horizon of the black hole as functions
of the modified gravity parameter $\alpha$.

####  III.1.2 Solution with Charge

We now add in (25) a Maxwell term and the whole action reads

$S=\int
d^{4}x\sqrt{-g}\left(\cfrac{f(R)}{2}-\cfrac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\cfrac{1}{12}R\phi^{2}-V(\phi)-\cfrac{1}{4}F^{2}\right)~{},$
(53)

where

$\displaystyle F^{2}$ $\displaystyle=$ $\displaystyle
F^{\mu\nu}F_{\mu\nu}~{},$ (54) $\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}~{}.$ (55)

By variation with respect to the inverse metric tensor, the scalar field and
the $U(1)$ field we obtain the Einstein equation (26), the Klein-Gordon
equation (27) and the Maxwell equation

$\nabla_{\mu}F^{\mu\nu}=0~{},$ (56)

while the energy-momentum tensor is now

$T^{total}_{\mu\nu}=T_{\mu\nu}^{\phi}+T^{EM}_{\mu\nu}~{},$ (57)

where

$T^{EM}_{\mu\nu}=F_{\mu\alpha}F^{\alpha}_{\nu}-\cfrac{1}{4}g_{\mu\nu}F^{2}~{}.$
(58)

Imposing the same metric ansatz (6) and the following ansatz for the
electromagnetic field, allowing only radial electric fields

$A_{\mu}=(\mathcal{A}(r),0,0,0)~{},$ (59)

we get

$-\frac{2\mathcal{A}^{\prime}(r)}{r}-\mathcal{A}^{\prime\prime}(r)=0\rightarrow\mathcal{A}(r)=\cfrac{Q}{r}~{},$
(60)

where $Q$ is the charge of the black hole.

Now, solving Einstein equations and (32) the only functions that change are

$\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{6-64\alpha^{2}Q^{2}}}{(4\alpha r-1)}~{},$ (61)
$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}\left(32\alpha^{2}Q^{2}-3\right)}{3(1-4\alpha
r)^{4}}~{},$ (62) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}\phi^{4}}{96\alpha^{2}Q^{2}-9}~{},$ (63)

while all other functions remain the same. For the vanishing of $Q$ the
solution turns back to the uncharged one. The electric charge does not appear
in the metric function but modifies the scalar field and scalar potential,
which is exactly what happens in the GR case [1, 3].

In order to have a real valued scalar field we should impose the condition

$-\frac{1}{4\alpha}\sqrt{\frac{3}{2}}<Q<\frac{1}{4\alpha}\sqrt{\frac{3}{2}}~{}.$
(64)

The scalar field still remains finite at the black hole horizon and now takes
the value

$\phi(r_{+})=-\frac{6\sqrt{6-64\alpha^{2}Q^{2}}}{\sqrt{10}+2}~{}.$ (65)

Since the metric function remains unchanged, the divergence of the scalar
field lies in between of the inner and event horizon of the black hole.

Thermodynamically, the temperature will be still given by equation (51), since
the electric charge does not appear in the metric function. The entropy will
change though, now the electric charge has an impact on the profile of the
scalar field. The entropy at the event horizon of the black hole will be given
by (50)

$S(r_{+})=\frac{2}{3}\left(11-2\sqrt{10}\right)\pi
Q^{2}+\frac{5\left(191\sqrt{10}-848\right)\pi}{6912\alpha^{2}}~{},$ (66)

where we have taken into account that the event horizon is located at
$r_{+}=(8+\sqrt{10})/24\alpha$. The second term in the relation (66) is the
entropy of the uncharged black hole. It is clear that the addition of the
electric charge gains more entropy for the black hole throught the scalar
field. The entropy is positive when the electric charge and the modified
gravity parameter $\alpha$ are related throught the inequalities

$\frac{\sqrt{10}}{\alpha}\sqrt{\frac{191\sqrt{10}-848}{\left(2\sqrt{10}-11\right)}}+96Q<0\hskip
14.22636pt\text{or}\hskip
14.22636pt\frac{\sqrt{10}}{\alpha}\sqrt{\frac{191\sqrt{10}-848}{\left(2\sqrt{10}-11\right)}}<96Q\hskip
5.69046pt\&\hskip 5.69046pt\frac{\sqrt{6}}{\alpha}>8Q~{}.$ (67)

Figure 6: The Bekenstein-Hawking $S(r_{+})$ entropy of the charged solution at
the event horizon of the black hole as a function of $\alpha$ while changing
the electric charge $Q$.

In FIG. 6 we give a plot of the entropy at the event horizon of the black hole
as a function of the modified gravity parameter $\alpha$ while changing the
electric charge. We can see that when the above relations hold, the entropy
becomes positive while otherwise is negative.

###  III.2 $f(R)=R-2\Lambda-2\alpha\sqrt{R-4\Lambda}$

####  III.2.1 Neutral Solution

In this Section we introduce a cosmological constant in the $f(R)$ function

$f(R)=R-2\Lambda-2\alpha\sqrt{R-4\Lambda}~{}.$ (68)

In this case the solution to the system of equations (29), (30), (31) and (32)
yields the following configurations

$\displaystyle f_{R}(r)$ $\displaystyle=$ $\displaystyle 1-\alpha r~{},$ (69)
$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{3\alpha
r}+\frac{3}{64\alpha^{2}r^{2}}-\frac{\Lambda r^{2}}{3},$ (70) $\displaystyle
f(r)$ $\displaystyle=$ $\displaystyle\frac{2\Lambda r^{2}-2\alpha
r+1}{r^{2}}~{},$ (71) $\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{6}}{1-4\alpha r}~{},$ (72) $\displaystyle V(r)$
$\displaystyle=$ $\displaystyle-\frac{4\alpha^{2}+\Lambda}{(4\alpha
r-1)^{4}}~{},$ (73) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle-\frac{1}{36}\phi^{4}\left(4\alpha^{2}+\Lambda\right)~{},$ (74)
$\displaystyle R(r)$ $\displaystyle=$ $\displaystyle
4\Lambda+\frac{1}{r^{2}}~{}.$ (75)

The trace of the energy-momentum tensor vanishes, meaning that the scalar
field is conformally coupled to gravity and the resulting potential preserves
the conformal invariance. The second derivative of the gravitational model
should be positive in order to be stable and for the choice of the $f(R)$
function (68), we have

$f_{RR}=\frac{\alpha}{2(R-4\Lambda)^{3/2}}~{},$ (76)

which is always positive for $\alpha>0$. The horizons can be obtained
analytically. For completeness we give the posssibly positive roots of the
metric function which for appropriate relations between the constants
represent black hole horizons.

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\sqrt{\frac{\alpha^{2}\left(\Lambda\sqrt[3]{L}+2\right)^{2}-3\Lambda}{\alpha^{2}\Lambda^{2}\sqrt[3]{L}}}-2\sqrt{\frac{2}{\Lambda}+\sqrt[3]{L}\left(-\frac{4}{\alpha\Lambda\sqrt[3]{L}\sqrt{\frac{\alpha^{2}\left(\Lambda\sqrt[3]{L}+2\right)^{2}-3\Lambda}{\alpha^{2}\Lambda^{2}\sqrt[3]{L}}}}-\frac{1}{4}\right)+\frac{\frac{3\Lambda}{\alpha^{2}}-4}{4\Lambda^{2}\sqrt[3]{L}}}\right)~{},$
(77) $\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(2\sqrt{\frac{2}{\Lambda}+\sqrt[3]{L}\left(\frac{4}{\alpha\Lambda\sqrt[3]{L}\sqrt{\frac{\alpha^{2}\left(\Lambda\sqrt[3]{L}+2\right)^{2}-3\Lambda}{\alpha^{2}\Lambda^{2}\sqrt[3]{L}}}}-\frac{1}{4}\right)+\frac{\frac{3\Lambda}{\alpha^{2}}-4}{4\Lambda^{2}\sqrt[3]{L}}}-\sqrt{\frac{\alpha^{2}\left(\Lambda\sqrt[3]{L}+2\right)^{2}-3\Lambda}{\alpha^{2}\Lambda^{2}\sqrt[3]{L}}}\right)~{},$
(78) $\displaystyle r_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\sqrt{\frac{\alpha^{2}\left(\Lambda\sqrt[3]{L}+2\right)^{2}-3\Lambda}{\alpha^{2}\Lambda^{2}\sqrt[3]{L}}}+2\sqrt{\frac{2}{\Lambda}+\sqrt[3]{L}\left(-\frac{4}{\alpha\Lambda\sqrt[3]{L}\sqrt{\frac{\alpha^{2}\left(\Lambda\sqrt[3]{L}+2\right)^{2}-3\Lambda}{\alpha^{2}\Lambda^{2}\sqrt[3]{L}}}}-\frac{1}{4}\right)+\frac{\frac{3\Lambda}{\alpha^{2}}-4}{4\Lambda^{2}\sqrt[3]{L}}}\right)~{},$
(79)

where

$L=\sqrt{\frac{88\alpha^{2}\Lambda-80\alpha^{4}+27\Lambda^{2}}{\alpha^{6}\Lambda^{5}}}+\frac{14}{\alpha^{2}\Lambda^{2}}-\frac{8}{\Lambda^{3}}~{}.$
(80)

In FIG. 7 we plot the metric function (70). For appropriate relations between
the parameter $\alpha$ and the cosmological constant, the positive
cosmological constant case gives dS spacetimes with two black hole horizons
and one cosmological horizon, which is what happens in the GR case [3]. As in
the asymptotically flat case, the divergence of the scalar field is in between
of the two black hole horizons. We compute the Kretschmann scalar for the
metric (70)

$\displaystyle R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}$
$\displaystyle=$
$\displaystyle\frac{8\Lambda^{2}}{3}+\frac{55}{48\alpha^{2}r^{6}}-\frac{3}{4\alpha^{3}r^{7}}+\frac{63}{512\alpha^{4}r^{8}}+\frac{4}{3\alpha
r^{5}}+\frac{4\Lambda}{3r^{2}}+\frac{1}{r^{4}}~{},$ (81)
$\displaystyle\lim_{r\rightarrow
0}R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}$
$\displaystyle\rightarrow$ $\displaystyle\infty~{}.$ (82)

The square of the Weyl tensor remains the same (48) as expected, since the
cosmological constant does not appear in the Weyl tensor.

To study thermodynamics we consider that the black hole horizon for the metric
(70) is the largest positive root of the metric function

$r_{+}=\text{Root}\left(9-32\alpha r\left(2\alpha\Lambda r^{3}-3\alpha
r+2\right)\right)~{}.$ (83)

Now using the relations (49), (50) we compute the Hawking temperature and the
Bekenstein-Hawking entropy as

$\displaystyle T(r_{+})$ $\displaystyle=$ $\displaystyle\frac{32\alpha
r_{+}\left(1-2\alpha\Lambda
r_{+}^{3}\right)-9}{384\pi\alpha^{2}r_{+}^{3}}~{},$ (84) $\displaystyle
S(r_{+})$ $\displaystyle=$ $\displaystyle\pi r_{+}^{2}\left(+1-\alpha
r_{+}-\frac{1}{\left(1-4\alpha r_{+}\right){}^{2}}\right)~{}.$ (85)

In order to make sure that $r_{+}$ represents the black hole horizon we solve
the metric function for the modified gravity parameter $\alpha$ and we obtain

$b(r_{+})=0\to\alpha_{1,2}=\frac{-8r_{+}\pm\sqrt{2}\sqrt{18\Lambda
r_{+}^{4}+5r_{+}^{2}}}{8\left(2\Lambda r_{+}^{4}-3r_{+}^{2}\right)}~{},$ (86)

where the $\alpha_{1}$ represents the solution with the plus sign. The
modified gravity parameter is positive and therefore both roots of the above
equation should be positive. For dS spacetimes $\alpha_{1,2}$ are both
positive when the cosmological constant satisfies
$0<\Lambda<\frac{3}{2r_{+}^{2}}$. For AdS spacetimes, $\alpha_{1,2}$ are both
positive when the cosmological constant satisfies
$-\frac{5}{18r_{+}^{2}}\leq\Lambda<0$. Now, we substitute the expressions for
$\alpha$ back to the temperature and we have

$T(r_{+})_{1,2}=-\frac{36\Lambda r_{+}^{3}\pm
4\sqrt{2}\sqrt{r_{+}^{2}\left(18\Lambda r_{+}^{2}+5\right)}+5r_{+}}{108\pi
r_{+}^{2}}~{}.$ (87)

For AdS spacetimes, the above temperature is positive when the cosmological
constant and the horizon satisfy
$-\frac{5}{18r_{+}^{2}}\leq\Lambda<-\frac{1}{4r_{+}^{2}}$ which also gives a
positive $\alpha$.

For dS spacetimes, the temperature is positive for $\alpha_{2}$ when the
cosmological constant satisfies: $0<\Lambda<\frac{5}{12r_{+}^{2}}$ and is
negative for $\alpha_{1}$.

From equation (85) we can see that regardless of the sign and value of the
cosmological constant, the entropy is always negative, for any
$r_{+}>0,\alpha>0$.

Figure 7: The metric function $b(r)$ while changing the cosmological constant,
where we’ve set $\alpha=1$. The black vertical line represents the position of
the divergence of the scalar field. The left plot represents dS spacetimes,
while the right AdS spacetimes.

####  III.2.2 Solution with Charge

We have seen in the asymptotically flat case that the addition of the Maxwell
term in the action, results in positive entropy when the constants of our
solution satisfy particular relations. Therefore we consider the action (53)
and the gravitational model (68) and solving the Einstein-Klein-Gordon-Maxwell
system we find that the only functions that change are

$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle\frac{\left(4\alpha^{2}+\Lambda\right)\left(32\alpha^{2}Q^{2}-3\right)}{3(1-4\alpha
r)^{4}}~{},$ (88) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{\phi^{4}\left(4\alpha^{2}+\Lambda\right)}{384\alpha^{2}Q^{2}-36}~{},$
(89)

while the scalar field is now given by (61) and the metric function, the Ricci
scalar and the functions $f_{R}$ and $f(r)$ are given by (70), (75), (71) and
(69) respectively. The temperature of the solution is given by (84) and we
discussed in detail the possibillity of positive temperature, while the
entropy is given by

$S(r_{+})=\pi r_{+}^{2}\left(1-\alpha
r_{+}+\frac{32\alpha^{2}Q^{2}-3}{3(1-4\alpha r_{+})^{2}}\right)~{}.$ (90)

We will now follow the same procedure as in the uncharded case. In order to
ensure that the entropy (90) is indeed the entropy at the black hole horizon
we solve the metric function for the parameter $\alpha$ and substitute back to
the entropy. We find that for particular range of values for the electric
charge and the cosmological constant, the entropy is positive for positive
$\alpha$. The inequalities are complicated and therefore, we will only give
the simplest one in order to illustrate the fact that we can have positive
entropy. For $\alpha_{2}$ and $-\frac{5}{18r_{+}^{2}}\leq\Lambda<0$ (AdS
spacetime) the entropy is positive when the electric charge and the black hole
horizon are related through

$4Q+\sqrt{3}\sqrt{\frac{r_{+}^{2}\left(5\left(\sqrt{36\Lambda
r_{+}^{2}+10}-4\right)-4\Lambda r_{+}^{2}\left(\sqrt{36\Lambda
r_{+}^{2}+10}-2\right)\right)}{2\Lambda r_{+}^{2}-3}}<0~{},$ (91)

or,

$4Q>\sqrt{3}\sqrt{\frac{r_{+}^{2}\left(5\left(\sqrt{36\Lambda
r_{+}^{2}+10}-4\right)-4\Lambda r_{+}^{2}\left(\sqrt{36\Lambda
r_{+}^{2}+10}-2\right)\right)}{2\Lambda r_{+}^{2}-3}}~{}.$ (92)

For these particular values we also have positive temperature.

##  IV Conclusions

In this work we studied $f(R)$ gravity theories in the presence of matter. We
considered a scalar field non-minimally coupled to gravity in the context of
$f(R)$ theories. We derived asymptotically flat or (A)dS exact black hole
solutions with dynamic Ricci curvature. We also studied the thermodynamics of
these black hole solutions. We calculated the temperature and the entropy and
because of the presence of the non-minimal coupling between the scalar field
and the scalar curvature the area law of the Bekenstein-Hawking entropy is
modified resulting to some interesting properties for the entropy.

We first considered the case where $f(R)=R-2\alpha\sqrt{R}$. The parameter
$\alpha$ which has the dimension of inverse length, introduces a non-linear
correction term to the Ricci scalar $R$. This parameter induces a charge in
the scalar field function which however is not independent of the black hole
mass and therefore cannot be detected asymptotically, failing in this way to
dress the black hole with a primary hair. Calculating the temperature of the
black hole solution we find that it is positive for any value of the parameter
$\alpha$. However, the entropy, having a contribution from the scalar field,
is negative regardless of the value of the parameter $\alpha$. To cure this
problem we introduced an electromagnetic field. Then we found that because of
the conformal invariance, the electric charge does not appear in the metric
function giving charge to the black hole solution, it appears however in the
scalar field function making the entropy positive.

We then introduced a cosmological constant in the $f(R)$ function
$f(R)=R-2\Lambda-2\alpha\sqrt{R-4\Lambda}$. We found dS and AdS black hole
solutions depending on the sign of the cosmological constant, while the scalar
field has the same behavior with the asymptotically flat case, it cannot give
hair to the black hole solutions. If we introduce electric charge this charge
appears in the scalar function and the interplay between the cosmological
constant and the electric charge can result to a positive entropy.

It is known that in order to have a black hole with a non-minimally coupled
scalar field regular at the horizon, a scale has to be introduced in the
theory [3, 62, 63], which, so far, has been considered to be a cosmological
constant. In this work we showed that this not the only possibility. A scale
in the form of a non-linear correction term in the gravitational part of the
action, also results to a regular scalar field at the black hole event
horizon. This would be interesting to be further investigated. One could also
consider non-linear electrodynamics, instead of Maxwell electrodynamics to see
if it is possible to construct regular black hole solutions (without a
singularity). In our case, in order to respect the conformal invariance, the
$f(R)$ model and the Ricci scalar are fixed in a particular way and regular
black hole solutions cannot be found since Ricci scalar is divergent at the
origin and so are the other curvature scalars. The introduction of non-linear
electrodynamics will break the conformal invariance making it hopeful to
construct regular black holes as it was done in [64, 65].

It would be interesting to extent this work to the study of other compact
objects in the $f(R)$ gravity theories. The conformally coupled scalar field
can provide the matter content of these theories. Then the astrophysical
observations in this strong gravity regime may give important information on
the departure from GR restricting the parameter $\alpha$ which expresses the
deviation from the Ricci scalar $R$.

## References

  * [1] N. Bocharova, K. Bronnikov and V. Melnikov, Vestn. Mosk. Univ. Fiz. Astron. 6, 706 (1970);
J. D. Bekenstein, Annals Phys. 82, 535 (1974);
J. D. Bekenstein, “Black Holes With Scalar Charge,” Annals Phys. 91, 75
(1975).

  * [2] K. A. Bronnikov and Y. N. Kireyev, “Instability of black holes with scalar charge,” Phys. Lett. A 67, 95 (1978).
  * [3] C. Martinez, R. Troncoso and J. Zanelli, “De Sitter black hole with a conformally coupled scalar field in four-dimensions,” Phys. Rev. D 67, 024008 (2003) [arXiv:hep-th/0205319 [hep-th]].
  * [4] T. J. T. Harper, P. A. Thomas, E. Winstanley and P. M. Young, “Instability of a four-dimensional de Sitter black hole with a conformally coupled scalar field,” Phys. Rev. D 70, 064023 (2004) [arXiv:gr-qc/0312104 [gr-qc]].
  * [5] C. Martinez, R. Troncoso and J. Zanelli, “Exact black hole solution with a minimally coupled scalar field,” Phys. Rev. D 70, 084035 (2004) [arXiv:hep-th/0406111 [hep-th]].
  * [6] C. Martinez and R. Troncoso, “Electrically charged black hole with scalar hair,” Phys. Rev. D 74, 064007 (2006) [arXiv:hep-th/0606130 [hep-th]].
  * [7] T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, “A New Class of Exact Hairy Black Hole Solutions,” Gen. Rel. Grav. 43, 163-180 (2011) [arXiv:0911.1711 [hep-th]].
  * [8] P. A. González, E. Papantonopoulos, J. Saavedra and Y. Vásquez, “Four-Dimensional Asymptotically AdS Black Holes with Scalar Hair,” JHEP 12, 021 (2013) [arXiv:1309.2161 [gr-qc]].
  * [9] P. A. González, E. Papantonopoulos, J. Saavedra and Y. Vásquez, “Extremal Hairy Black Holes,” JHEP 11, 011 (2014) [arXiv:1408.7009 [gr-qc]].
  * [10] A. Anabalon and A. Cisterna, “Asymptotically (anti) de Sitter Black Holes and Wormholes with a Self Interacting Scalar Field in Four Dimensions,” Phys. Rev. D 85, 084035 (2012) [arXiv:1201.2008 [hep-th]].
  * [11] T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, “Phase Transition to a Hairy Black Hole in Asymptotically Flat Spacetime,” JHEP 11, 133 (2013) [arXiv:1308.5280 [hep-th]].
  * [12] T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, “Scalar Hair from a Derivative Coupling of a Scalar Field to the Einstein Tensor,” Class. Quant. Grav. 29, 205011 (2012) [arXiv:1111.0263 [gr-qc]].
  * [13] M. Rinaldi, “Black holes with non-minimal derivative coupling,” Phys. Rev. D 86, 084048 (2012) [arXiv:1208.0103 [gr-qc]].
  * [14] M. Minamitsuji, “Solutions in the scalar-tensor theory with nonminimal derivative coupling,” Phys. Rev. D 89, 064017 (2014) [arXiv:1312.3759 [gr-qc]].
  * [15] A. Anabalon, A. Cisterna and J. Oliva, “Asymptotically locally AdS and flat black holes in Horndeski theory,” Phys. Rev. D 89, 084050 (2014) [arXiv:1312.3597 [gr-qc]].
  * [16] S. Boudet, F. Bombacigno, G. Montani and M. Rinaldi, “Superentropic black hole with Immirzi hair,” Phys. Rev. D 103, no.8, 084034 (2021) doi:10.1103/PhysRevD.103.084034 [arXiv:2012.02700 [gr-qc]].
  * [17] A. Cisterna, A. Neira-Gallegos, J. Oliva and S. C. Rebolledo-Caceres, “Plebański-Demiański solutions in quadratic gravity with conformally coupled scalar fields,” Phys. Rev. D 103, no.6, 064050 (2021) doi:10.1103/PhysRevD.103.064050 [arXiv:2101.03628 [gr-qc]].
  * [18] M. Bravo-Gaete, C. G. Gaete, L. Guajardo and S. G. Rodríguez, “Towards the emergence of nonzero thermodynamical quantities for Lanczos-Lovelock black holes dressed with a scalar field,” Phys. Rev. D 104, no.4, 044027 (2021) doi:10.1103/PhysRevD.104.044027 [arXiv:2103.15634 [gr-qc]].
  * [19] C. Erices, P. Filis and E. Papantonopoulos, “Hairy black holes in disformal scalar-tensor gravity theories,” Phys. Rev. D 104, no.2, 024031 (2021) doi:10.1103/PhysRevD.104.024031 [arXiv:2104.05644 [gr-qc]].
  * [20] A. De Felice and S. Tsujikawa,“f(R) theories,” Living Rev. Rel. 13, 3 (2010) [arXiv:1002.4928 [gr-qc]].
  * [21] T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys. 82, 451-497 (2010) [arXiv:0805.1726 [gr-qc]].
  * [22] L. Sebastiani and S. Zerbini, “Static Spherically Symmetric Solutions in F(R) Gravity,” Eur. Phys. J. C 71, 1591 (2011) [arXiv:1012.5230 [gr-qc]].
  * [23] T. Multamaki and I. Vilja, “Spherically symmetric solutions of modified field equations in f(R) theories of gravity,” Phys. Rev. D 74, 064022 (2006) [arXiv:astro-ph/0606373 [astro-ph]].
  * [24] S. H. Hendi, “(2+1)-Dimensional Solutions in $F(R)$ Gravity,” Int. J. Theor. Phys. 53, no.12, 4170-4181 (2014) [arXiv:1410.7527 [gr-qc]].
  * [25] S. H. Hendi, B. Eslam Panah and R. Saffari, “Exact solutions of three-dimensional black holes: Einstein gravity versus $F(R)$ gravity,” Int. J. Mod. Phys. D 23, no.11, 1450088 (2014) [arXiv:1408.5570 [hep-th]].
  * [26] G. G. L. Nashed and S. Nojiri, “Non-trivial black hole solutions in $\mathit{f(R)}$ gravitational theory,” Phys. Rev. D 102, 124022 (2020) doi:10.1103/PhysRevD.102.124022 [arXiv:2012.05711 [gr-qc]].
  * [27] E. Elizalde, G. G. L. Nashed, S. Nojiri and S. D. Odintsov, “Spherically symmetric black holes with electric and magnetic charge in extended gravity: physical properties, causal structure, and stability analysis in Einstein’s and Jordan’s frames,” Eur. Phys. J. C 80, no.2, 109 (2020) [arXiv:2001.11357 [gr-qc]].
  * [28] G. G. L. Nashed and S. Nojiri, “Analytic charged BHs in f(R) gravity,” Phys. Lett. B 820, 136475 (2021) doi:10.1016/j.physletb.2021.136475 [arXiv:2010.04701 [hep-th]].
  * [29] G. G. L. Nashed and K. Bamba, “Charged spherically symmetric Taub–NUT black hole solutions in $f(R)$ gravity,” PTEP 2020, no.4, 043E05 (2020) [arXiv:1902.08020 [gr-qc]].
  * [30] Z. Amirabi, M. Halilsoy and S. Habib Mazharimousavi, “Generation of spherically symmetric metrics in f(R) gravity,” Eur. Phys. J. C 76, no.6, 338 (2016) doi:10.1140/epjc/s10052-016-4164-z [arXiv:1509.06967 [gr-qc]].
  * [31] G. G. L. Nashed and S. Capozziello, “Charged spherically symmetric black holes in $f(R)$ gravity and their stability analysis,” Phys. Rev. D 99, no.10, 104018 (2019) [arXiv:1902.06783 [gr-qc]].
  * [32] J. A. R. Cembranos, A. de la Cruz-Dombriz and P. Jimeno Romero, “Kerr-Newman black holes in $f(R)$ theories,” Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1450001 [arXiv:1109.4519 [gr-qc]].
  * [33] A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, “Black Holes in f(R) theories,” Phys. Rev. D 80 (2009) 124011 Erratum: [Phys. Rev. D 83 (2011) 029903] [arXiv:0907.3872 [gr-qc]].
  * [34] S. H. Hendi, B. Eslam Panah and S. M. Mousavi, “Some exact solutions of F(R) gravity with charged (a)dS black hole interpretation,” Gen. Rel. Grav. 44, 835 (2012) [arXiv:1102.0089 [hep-th]].
  * [35] Z. Y. Tang, B. Wang and E. Papantonopoulos, “Exact charged black hole solutions in $D$-dimensions in $f(R)$ gravity,” Eur. Phys. J. C 81, no.4, 346 (2021) [arXiv:1911.06988 [gr-qc]].
  * [36] S. Nojiri, S. D. Odintsov and V. Faraoni, “Searching for dynamical black holes in various theories of gravity,” Phys. Rev. D 103 (2021) no.4, 044055 [arXiv:2010.11790 [gr-qc]].
  * [37] Z. Y. Tang, B. Wang, T. Karakasis and E. Papantonopoulos, “Curvature scalarization of black holes in f(R) gravity,” Phys. Rev. D 104, no.6, 064017 (2021) doi:10.1103/PhysRevD.104.064017 [arXiv:2008.13318 [gr-qc]].
  * [38] T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “Black holes of (2+1)-dimensional $f(R)$ gravity coupled to a scalar field,” Phys. Rev. D 103, no.6, 064063 (2021) [arXiv:2101.06410 [gr-qc]].
  * [39] G. G. L. Nashed, “New rotating AdS/dS black holes in $f(R)$ gravity,” Phys. Lett. B 815, 136133 (2021) [arXiv:2102.11722 [gr-qc]].
  * [40] G. G. L. Nashed, “Uniqueness of non-trivial spherically symmetric black hole solution in special classes of F ( R ) gravitational theory,” Phys. Lett. B 812, 136012 (2021) [arXiv:2101.02205 [gr-qc]].
  * [41] K. V. Staykov, D. D. Doneva, S. S. Yazadjiev and K. D. Kokkotas, “Gravitational wave asteroseismology of neutron and strange stars in R2 gravity,” Phys. Rev. D 92, no.4, 043009 (2015) [arXiv:1503.04711 [gr-qc]].
  * [42] D. D. Doneva, S. S. Yazadjiev and K. D. Kokkotas, “The I-Q relations for rapidly rotating neutron stars in $f(R)$ gravity,” Phys. Rev. D 92, no.6, 064015 (2015) [arXiv:1507.00378 [gr-qc]].
  * [43] S. S. Yazadjiev, D. D. Doneva, K. D. Kokkotas and K. V. Staykov, “Non-perturbative and self-consistent models of neutron stars in R-squared gravity,” JCAP 06, 003 (2014) [arXiv:1402.4469 [gr-qc]].
  * [44] F. Ozel, G. Baym and T. Guver, “Astrophysical Measurement of the Equation of State of Neutron Star Matter,” Phys. Rev. D 82, 101301 (2010) [arXiv:1002.3153 [astro-ph.HE]].
  * [45] A. S. Arapoglu, C. Deliduman and K. Y. Eksi, “Constraints on Perturbative f(R) Gravity via Neutron Stars,” JCAP 07, 020 (2011) [arXiv:1003.3179 [gr-qc]].
  * [46] K. V. Staykov, D. D. Doneva, S. S. Yazadjiev and K. D. Kokkotas, “Slowly rotating neutron and strange stars in $R^{2}$ gravity,” JCAP 10, 006 (2014) [arXiv:1407.2180 [gr-qc]].
  * [47] P. Cañate, “A no-hair theorem for black holes in $f(R)$ gravity,” Class. Quant. Grav. 35, no.2, 025018 (2018).
  * [48] J. Sultana and D. Kazanas, “A no-hair theorem for spherically symmetric black holes in $R^{2}$ gravity,” Gen. Rel. Grav. 50, no.11, 137 (2018) [arXiv:1810.02915 [gr-qc]].
  * [49] P. Cañate, L. G. Jaime and M. Salgado, “Spherically symmetric black holes in $f(R)$ gravity: Is geometric scalar hair supported ?,” Class. Quant. Grav. 33, no.15, 155005 (2016) [arXiv:1509.01664 [gr-qc]].
  * [50] S. Pi, Y. l. Zhang, Q. G. Huang and M. Sasaki, “Scalaron from $R^{2}$-gravity as a heavy field,” JCAP 05, 042 (2018) [arXiv:1712.09896 [astro-ph.CO]].
  * [51] A. de la Cruz-Dombriz, E. Elizalde, S. D. Odintsov and D. Sáez-Gómez, “Spotting deviations from R2 inflation,” JCAP 05, 060 (2016) [arXiv:1603.05537 [gr-qc]].
  * [52] S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,” Phys. Rept. 505 (2011), 59-144 [arXiv:1011.0544 [gr-qc]].
  * [53] S. Capozziello and M. De Laurentis, “Extended Theories of Gravity,” Phys. Rept. 509 (2011), 167-321 [arXiv:1108.6266 [gr-qc]].
  * [54] A. M. Barlow, D. Doherty and E. Winstanley, “Thermodynamics of de Sitter black holes with a conformally coupled scalar field,” Phys. Rev. D 72, 024008 (2005) [arXiv:gr-qc/0504087 [gr-qc]].
  * [55] E. Winstanley, “Classical and thermodynamical aspects of black holes with conformally coupled scalar field hair,” Conf. Proc. C 0405132, 305-323 (2004) [arXiv:gr-qc/0408046 [gr-qc]].
  * [56] O. B. Zaslavskii, “Thermodynamics of black holes with an infinite effective area of a horizon,” Class. Quant. Grav. 19, 3783-3798 (2002) [arXiv:gr-qc/0206018 [gr-qc]].
  * [57] C. A. R. Herdeiro and E. Radu, “Asymptotically flat black holes with scalar hair: a review,” Int. J. Mod. Phys. D 24, no.09, 1542014 (2015) [arXiv:1504.08209 [gr-qc]].
  * [58] A. Kehagias, C. Kounnas, D. Lüst and A. Riotto, “Black hole solutions in $R^{2}$ gravity,” JHEP 05, 143 (2015) [arXiv:1502.04192 [hep-th]].
  * [59] Y. Zheng and R. J. Yang, “Horizon thermodynamics in $f(R)$ theory,” Eur. Phys. J. C 78, no.8, 682 (2018) [arXiv:1806.09858 [gr-qc]].
  * [60] T. Damour and G. Esposito-Farese, Phys. Rev. Lett. 70, 2220-2223 (1993)
  * [61] H. Sotani and K. D. Kokkotas, Phys. Rev. D 70, 084026 (2004) [arXiv:gr-qc/0409066 [gr-qc]].
  * [62] M. Khodadi, A. Allahyari, S. Vagnozzi and D. F. Mota, “Black holes with scalar hair in light of the Event Horizon Telescope,” JCAP 09, 026 (2020) [arXiv:2005.05992 [gr-qc]].
  * [63] S. Priyadarshinee, S. Mahapatra and I. Banerjee, “Analytic topological hairy dyonic black holes and thermodynamics,” Phys. Rev. D 104, no.8, 084023 (2021) doi:10.1103/PhysRevD.104.084023 [arXiv:2108.02514 [hep-th]].
  * [64] M. E. Rodrigues, E. L. B. Junior, G. T. Marques and V. T. Zanchin, “Regular black holes in $f(R)$ gravity coupled to nonlinear electrodynamics,” Phys. Rev. D 94, no.2, 024062 (2016) [arXiv:1511.00569 [gr-qc]].
  * [65] M. E. Rodrigues, J. C. Fabris, E. L. B. Junior and G. T. Marques, “Generalisation for regular black holes on general relativity to $f(R)$ gravity,” Eur. Phys. J. C 76, no.5, 250 (2016) [arXiv:1601.00471 [gr-qc]].


# Breakdown of superconductivity in a magnetic field with self-intersecting
zero set

and  Kamel Attar

###### Abstract.

We prove that the lowest eigenvalue of the Laplace operator with a magnetic
field having a self-intersecting zero set is a monotone function of the
parameter defining the strength of the magnetic field, in a neighborhood of
infinity. We apply this monotonicity result on the study of the transition
from superconducting to normal states for the Ginzburg-Landau model, and prove
that the transition occurs at a unique threshold value of the applied magnetic
field.

##  1\. Introduction

###  1.1. Breakdown of superconductivity.

Superconductivity is a state of metals and alloys that appear below a certain
critical temperature $\displaystyle T_{c}$. When the temperature is below
$\displaystyle T_{c}$ the body must be in the superconducting state; if an
applied magnetic field (with small intensity) attempts to destroy the
superconductivity, an induced magnetic field appears and repels the applied
magnetic field. If the intensity of the applied magnetic field is increased
gradually past a critical value, the superconductor can no longer resist the
magnetic field and the appearance of superconductivity in the metal decreases
gradually past critical conducting states (superconducting state, mixed
conducting state and normal conducting state). Eventually, a large magnetic
field destroys superconductivity from the sample (see Figure 1).

Figure 1. Schematic representation of a material shows the difference states.

In the case of a uniform applied magnetic field, phase transitions associated
with a type II superconductor are marked by three thresholds:

  * •

$\displaystyle H_{C_{1}}$ is the intensity of the magnetic field where a
superconductor switches from the superconducting state to the mixed conducting
state (vortex state). In this case the superconductor allows the applied
magnetic field to pass through small regions of the sample [31].

  * •

$\displaystyle H_{C_{2}}$ is the intensity of the magnetic field that
corresponds to the transition from superconductivity located in regions far
from the boundary to surface superconductivity [29, 32, 15].

  * •

$\displaystyle H_{C_{3}}$ is the intensity of the magnetic field that
corresponds to the transition from surface superconductivity to the normal
conducting state; in this case the superconductivity can no longer exist in
the superconductor [29, 10, 7].

Non uniform magnetic fields is the subject of recent research:

  * •

Non-vanishing smooth magnetic fields [10, 30, 25] ;

  * •

Sign-changing smooth magnetic fields [28, 18] ;

  * •

Non-smooth magnetic fields [1, 3, 23] .

In this contribution, we study smooth magnetic fields with a self-intersecting
zero set.

###  1.2. Magnetic field with self-intersecting zeros

Suppose that $\displaystyle\Omega\subset\mathbb{R}^{2}$ is bounded, open,
simply connected with smooth boundary. Consider a vertical magnetic field of
the form

$\displaystyle B_{0}\vec{z}$

where the function $\displaystyle B_{0}\in C^{\infty}(\overline{\Omega})$ has
a non-trivial zero set

$\Gamma=\\{x\in\overline{\Omega}:B_{0}(x)=0\\}\qquad\text{and}\qquad\Sigma=\\{x\in\Gamma:\nabla
B_{0}(x)=0\\}$ (1.1)

and satisfies the following properties (see [8])

$\displaystyle\Gamma\neq\varnothing,\quad\Sigma\neq\varnothing{\rm\leavevmode\nobreak\
is\leavevmode\nobreak\ finite,}\quad\Sigma\cap\partial\Omega=\varnothing$

and

$\left\\{\begin{array}[]{rll}|B_{0}|+|\nabla B_{0}|>0&\mbox{ in
}\overline{\Omega}\setminus\Sigma&(a)\\\ \nabla B_{0}\times\vec{n}\neq
0&\mbox{ on }\Gamma\cap\partial\Omega&(b)\\\ |\nabla B_{0}|+|{\rm
Hess}B_{0}|>0&\mbox{ on }\Gamma&(c)\\\ \vec{\tau_{1}}\times\vec{\tau_{2}}\neq
0&\mbox{ on }\Sigma&(d)\,.\end{array}\right.$ (1.2)

Here $\displaystyle\vec{n}$ is the unit normal vector to the boundary,
$\displaystyle\vec{\tau}_{1}$ and $\displaystyle\vec{\tau}_{2}$ are the unit
tangent vector on the intersection point $\displaystyle x\in\Sigma$, and
$\displaystyle{\rm Hess}B_{0}$ is the Hessian matrix of the magnetic field at
the point $\displaystyle x$ which has two non-zero eigenvalues
$\displaystyle\lambda^{\rm Hess}_{1}(x)$ and $\displaystyle\lambda^{\rm
Hess}_{2}(x)$ with opposite signs and labeled as follows

$|\lambda^{\rm Hess}_{1}(x)|\leq|\lambda^{\rm
Hess}_{2}(x)|\quad(x\in\Sigma)\,.$ (1.3)

Assumption $\displaystyle\eqref{B(x)}_{(a)}$ implies that for any open set
$\displaystyle\omega$ relatively compact in $\displaystyle\Omega$,
$\displaystyle\Gamma\cap\omega$ is either empty, or consists of a union of
smooth curves and the quantity $\displaystyle|\nabla B_{0}|$ does not vanish
on $\displaystyle\Gamma$. We see from $\displaystyle\eqref{B(x)}_{(b)}$ that
the $\displaystyle B_{0}$ is allowed to vanish on a finite number of boundary
points and that the curve $\displaystyle\Gamma$ cannot intersect tangentially
$\displaystyle\partial\Omega$. Assumptions $\displaystyle\eqref{B(x)}_{(c)}$
and $\displaystyle\eqref{B(x)}_{(d)}$ tell us that the smooth curves can
intersect on isolated points where $\displaystyle|\nabla B_{0}|$ vanishes.

The case where the magnetic field vanishes non-degenerately along a smooth non
self intersecting curve, is the subject of numerous works in the contexts of
shape optimization[27], superconductivity [28, 18, 2] and semiclassical
spectral asymptotics [9, 6].

Figure 2. Schematic representation of a domain subject to a variable magnetic
field that vanishes along the curve $\displaystyle\Gamma$ which crosses itself
inside the domain.

###  1.3. Magnetic Laplacian and strong diamagnetism

Consider the unique vector field $\displaystyle\mathbf{F}\in
H^{1}(\Omega;\mathbb{R}^{2})$ satisfying the following properties

$\nu\cdot\mathbf{F}=0\leavevmode\nobreak\ {\rm on\leavevmode\nobreak\
}\partial\Omega,\quad{\rm div}\mathbf{F}=0\leavevmode\nobreak\
\&\leavevmode\nobreak\
\operatorname{curl}\mathbf{F}=B_{0}{\leavevmode\nobreak\ \rm
in\leavevmode\nobreak\ }\Omega\,,$ (1.4)

where $\displaystyle\nu$ is the unit outward normal of the boundary of
$\displaystyle\Omega$.

Let us introduce the magnetic Schrödinger operator

$P_{b\mathbf{F}}^{\Omega}=-(\nabla-ib\mathbf{F})^{2}\,,$ (1.5)

with domain

$\displaystyle
D(P_{b\mathbf{F}}^{\Omega}):=\\{u\in\mathcal{V}(\Omega)\leavevmode\nobreak\
:\leavevmode\nobreak\ P_{b\mathbf{F}}^{\Omega}u\in
L^{2}(\Omega)\leavevmode\nobreak\ \&\leavevmode\nobreak\ \nu\cdot(\nabla-
ib\mathbf{F})u=0\leavevmode\nobreak\ {\rm on}\leavevmode\nobreak\
\partial\Omega\\}\,.$

The lowest eigenvalue (ground state energy) of this operator is

$\mu_{1}(b\mathbf{F})=\inf_{\begin{subarray}{c}\phi\in H^{1}(\Omega)\\\
\phi\neq
0\end{subarray}}\left(\frac{\mathcal{Q}_{b\mathbf{F}}^{\Omega}(\phi)}{\|\phi\|^{2}_{L^{2}(\Omega)}}\right)\,,$
(1.6)

where

$\mathcal{Q}_{b\mathbf{F}}^{\Omega}(\phi)=\int_{\Omega}|(\nabla-
ib\mathbf{F})\phi|^{2}\,dx\,.$ (1.7)

The large field limit $\displaystyle b\to+\infty$ can be transformed to a
semi-classical limit by introducing the small parameter $\displaystyle
h=\frac{1}{b}$. The full asymptotic expansion of the lowest eigenvalue
$\displaystyle P_{b\mathbf{F}}^{\Omega}$ is derived by Dauge, Miqueu and
Raymond [8]. Based on it, we prove the monotonicity of
$\displaystyle\mu_{1}(b\mathbf{F})$ with respect to $\displaystyle b$ in a
neighborhood of $\displaystyle+\infty$. This property is named _strong
diamagnetism_ in the literature [10, 11, 12].

######  Theorem 1.1.

There exists $\displaystyle b_{0}>0$, such that the function $\displaystyle
b\longmapsto\mu_{1}(b\mathbf{F})$ is monotone increasing on
$\displaystyle[b_{0},+\infty[$.

Strong diamagnetism has been proved earlier under different assumptions on
$\displaystyle\Omega$ and $\displaystyle B_{0}$ [11, 5, 4]. Counterexamples of
strong diamagnetism do exist [16, 19, 21, 22].

###  1.4. Ginzburg-Landau model

Consider the Ginzburg-Landau functional

$\mathcal{E}_{\kappa,H,B_{0}}(\psi,\mathbf{A})=\int_{\Omega}\left(|(\nabla-i\kappa
H\mathbf{A})\psi|^{2}+\frac{\kappa^{2}}{2}(1-|\psi|^{2})^{2}\right)\,dx+\kappa^{2}H^{2}\int_{\Omega}|\operatorname{curl}\mathbf{A}-B_{0}|^{2}\,dx\,.$
(1.8)

The vector field $\displaystyle\mathbf{A}$ is called the magnetic potential
which describes the induced magnetic field in the sample via

$\displaystyle\operatorname{curl}\mathbf{A}=\overrightarrow{\nabla}\times\mathbf{A}\,.$

The positive number $\displaystyle\kappa$ is the characteristic scale of the
superconductor that distinguishes the material of the sample; we will study
the case of materials of type II where $\displaystyle\kappa$ is sufficiently
large ($\displaystyle\kappa\to\infty$). The complex-valued function
$\displaystyle\psi$ is the order parameter that rises in the superconductor
phase, where the modulus squared $\displaystyle|\psi(x)|^{2}$ can be
interpreted as the local density of the superconducting electron Cooper pairs
as follows:

  * •

If $\displaystyle|\psi(x)|\neq 0$, then at location $\displaystyle x$, the
material is in the superconducting state.

  * •

If $\displaystyle|\psi(x)|=0$, then at location $\displaystyle x$, there
exists no Cooper pairs, and the material is in the normal state.

  * •

If $\displaystyle|\psi(x)|$ has zeros but does not vanish identically, the
material is in the mixed state.

Since the functional is invariant under gauge transformations
$\displaystyle(\psi,\mathbf{A})\to(e^{i\kappa
H\chi}\psi,\mathbf{A}-\nabla\chi)$, it is enough to consider configurations
$\displaystyle(\psi,\mathbf{A})$ in the space $\displaystyle
H^{1}(\Omega;\mathbb{C})\times H^{1}_{\operatorname{div}}(\Omega)$ of sobolev
functions, where

$H^{1}_{\operatorname{div}}(\Omega)=\Big{\\{}\mathbf{A}=(\mathbf{A}_{1},\mathbf{A}_{2})\in
H^{1}(\Omega)\times H^{1}(\Omega)\leavevmode\nobreak\ :\leavevmode\nobreak\
\operatorname{div}\mathbf{A}=0\leavevmode\nobreak\ {\rm
in}\leavevmode\nobreak\ \Omega\,,\,\mathbf{A}\cdot\nu=0\leavevmode\nobreak\
{\rm on}\,\partial\Omega\,\Big{\\}}\,,$ (1.9)

with $\displaystyle\nu$ being the unit interior normal vector of
$\displaystyle\partial\Omega$.

The equilibrium state of the system will be where the total energy
$\displaystyle\mathcal{E}_{\kappa,H,B_{0}}$ is minimal, we denote by
$\displaystyle(\psi,\mathbf{A})$ the minimizing configuration of
$\displaystyle\mathcal{E}_{\kappa,H,B_{0}}$. When $\displaystyle\psi=0$, the
minimizing configuration is in the form $\displaystyle(0,\mathbf{F})$ with
$\displaystyle\mathbf{F}$ in $\displaystyle
H^{1}_{\operatorname{div}}(\Omega)$ such that
$\displaystyle\operatorname{curl}\mathbf{F}=B_{0}$. Notice that this solution
is unique (up to a gauge transformation) and we call the pair
$\displaystyle(0,\mathbf{F})$ the normal state.

### Ginzburg-Landau equations

Minimizers, $\displaystyle(\psi,\mathbf{A})\in H^{1}(\Omega;\mathbb{C})\times
H^{1}_{\operatorname{div}}(\Omega)$ of (1.8) have to satisfy the Ginzburg-
Landau equations,

$\left\\{\begin{array}[]{llll}-(\nabla-i\kappa
H\mathbf{A})^{2}\psi=\kappa^{2}\,(1-|\psi|^{2})\,\psi&{\rm in}&\Omega&(a)\\\
-\nabla^{\bot}\operatorname{curl}(\mathbf{A}-\mathbf{F})=\displaystyle\frac{1}{\kappa
H}\operatorname{Im}(\overline{\psi}\,(\nabla-i\kappa H\mathbf{A})\psi)&{\rm
in}&\Omega&(b)\\\ \nu\cdot(\nabla-i\kappa H\mathbf{A})\psi=0&{\rm
on}&\partial\Omega&(c)\\\
\operatorname{curl}\mathbf{A}=\operatorname{curl}\mathbf{F}&{\rm
on}&\partial\Omega&(d)\,.\end{array}\right.$ (1.10)

Here,
$\displaystyle\nabla^{\bot}\operatorname{curl}\mathbf{A}=\Big{(}\partial_{x_{2}}(\operatorname{curl}\mathbf{A}),-\partial_{x_{1}}(\operatorname{curl}\mathbf{A})\Big{)}$.
The boundary conditions state that the supperconducting current can not flow
outside the sample (the sample is surrounded by vacuum).

A solution $\displaystyle(\psi,\mathbf{A})$ of (1.10) is said to be trivial if
$\displaystyle\psi=0$ everywhere (consequently
$\displaystyle\operatorname{curl}\mathbf{A}=b$ everywhere). Let us recall a
simple criterion [25, Lemma 2.1] for the existence of a non-trivial solution
of the Ginzburg-Landau equations in (1.10).

######  Proposition 1.2 (Sufficient condition for existence of non-trivial
solutions).

For all $\displaystyle\kappa$, $\displaystyle H>0$, if
$\displaystyle\mu_{1}(\kappa H\mathbf{F})<\kappa^{2}$, then every minimizer of
$\displaystyle\mathcal{E}_{\kappa,H,B_{0}}$ is non-trivial.

Our main result proves that the condition $\displaystyle\mu_{1}(\kappa
H\mathbf{F})<\kappa^{2}$ is also necessary, provided $\displaystyle\kappa$ is
sufficiently large.

######  Theorem 1.3 (Necessary condition for existence of non-trivial
solutions).

There exists $\displaystyle\kappa_{0}>0$ such that, for all
$\displaystyle\kappa\geq\kappa_{0}$ and $\displaystyle H>0$, the following two
properties are equivalent:

  * A.

There exists a solution $\displaystyle(\psi,\mathbf{A})$ of (1.10) such that
$\displaystyle\psi\not\equiv 0$ .

  * B.

$\displaystyle\mu_{1}(\kappa H\mathbf{F})<\kappa^{2}$.

For non sign-changing smooth magnetic fields and smooth domains, Theorem 1.3
was proved by Fournais-Helffer [13]. For uniform magnetic fields and non-
smooth domains, it was proved by [5].

### Discussion

Combining Theorems 1.1 and Theorem 1.3, we know that the transition from
trivial to non-trivial solution occurs at a unique threshold $\displaystyle
H_{C_{3}}(\kappa)$. More precisely,

$\displaystyle\exists\,\kappa_{0}>0,\leavevmode\nobreak\
\forall\,\kappa\geq\kappa_{0},\leavevmode\nobreak\
\exists\,H_{C_{3}}(\kappa)>0$

such that

  * •

$\displaystyle H=H_{C_{3}}(\kappa)$ is the unique solution of
$\displaystyle\mu_{1}(\kappa H)=\kappa^{2}$ ;

  * •

For $\displaystyle H<H_{C_{3}}(\kappa)$, every minimizer of
$\displaystyle\mathcal{E}_{\kappa,H,B_{0}}$ is non-trivial (i.e. the order
parameter does not vanish everywhere) ;

  * •

For $\displaystyle H\geq H_{C_{3}}(\kappa)$, every solution
$\displaystyle(\psi,\mathbf{A})$ of (1.10) is trivial (i.e.
$\displaystyle\psi\equiv 0$) .

The existing spectral asymptotics [8] for the eigenvalue
$\displaystyle\mu_{1}(b\mathbf{F})$ also yield a complete asymptotic expansion
of $\displaystyle H_{C_{3}}(\kappa)$; in fact (see Theorem 2.1)

$\displaystyle
H_{C_{3}}(\kappa)\sim\kappa^{3}\sup_{x\in\Sigma}\left(\frac{1}{|\lambda^{\rm
Hess}_{2}(x)|\left[\mathcal{X}_{1}\Big{(}\varepsilon(x)\Big{)}\right]^{2}}\right)+\sum\limits_{j=1}^{+\infty}c_{j}\kappa^{3-j}\qquad(\kappa\to+\infty)\,.$

Here

$\varepsilon(x)=\sqrt{\frac{|\lambda^{\rm Hess}_{1}(x)|}{|\lambda^{\rm
Hess}_{2}(x)|}},$ (1.11)

and for $\displaystyle\varepsilon>0$,
$\displaystyle\mathcal{X}_{1}(\varepsilon)$ denotes the first eigenvalue of
the following operator

$\mathcal{J}^{\varepsilon}:=D_{t}^{2}+\left(D_{s}-\left(\frac{t^{3}}{3}-\varepsilon^{2}s^{2}t\right)\right)^{2}\,,\qquad\text{in}\leavevmode\nobreak\
L^{2}(\mathbb{R}^{2})\,.$ (1.12)

Furthermore, for $\displaystyle\kappa\geq\kappa_{0}$ and $\displaystyle
H<H_{C_{3}}(\kappa)$, the local minimizers of the functional
$\displaystyle\mathcal{E}_{\kappa,H,B_{0}}$ are necessarily non-trivial. In
fact, $\displaystyle(0,\mathbf{F})$ is a local minimizer (locally stable)
provided that the second variation of the $\displaystyle\mathcal{E}$ is
positive,

$\displaystyle{\rm Hess}_{(0,\mathbf{F})}>0\,,$

which is equivalent to the property $\displaystyle\mu_{1}(\kappa
H\mathbf{F})>\kappa^{2}$.

### Oscillations and Little-Parks effect

The case studied in this paper is consistent with the generically observed
monotone transition between superconduting and normal phases. Equally
interesting are non-generic cases where the sample oscillates between the
superconducting and normal phases before setting definitely to the normal
state (Little-Parks effect [26], see Fig. 3). Breaking the monotonic
transition is viable by topological obstructions related to the domain or the
magnetic field. We refer to [16, 22, 19, 20, 21] for settings with
oscillations and to [13, 5, 14, 4] for generic monotone settings.

Figure 3. Schematic representation of a sample consistent with the Little-
Parks effect: Between $\displaystyle\overline{H}_{C_{3}}(\kappa)$ and
$\displaystyle\underline{H}_{C_{3}}(\kappa)$, the sample oscillates between
normal and superconducting phases.

### Organization of the paper

The rest of the paper consists of three sections: Sec. 2 is devoted to the
analysis of the monotonicity of $\displaystyle b\to\mu_{1}(b\mathbf{F})$; Sec.
3 establishes the breakdown of superconductivity under Assumption 1.2;
finally, Sec. 4 is devoted to the proof of Theorem 1.3 on the superconducting-
normal phase transition.

##  2\. Strong diamagnetism

In this section we prove Theorem 1.1 regarding the monotonicity of the
function $\displaystyle b\mapsto\mu_{1}(b\mathbf{F})$. We follow the approach
of Fournais-Helffer [11] whose ingredients are

  * •

The full asymptotic expansion for the lowest eigenvalue ;

  * •

Proposition 2.2 below.

The full asymptotic expansion for $\displaystyle\mu_{1}(b\mathbf{F})$ was
obtained in [8]; we recall this result below.

######  Theorem 2.1 (Dauge-Miqueu-Raymond [8]).

Let $\displaystyle\Omega\subset\mathbb{R}^{2}$ be an open bounded set with a
smooth boundary. Under Assumption 1.2. There exists a sequence
$\displaystyle\\{\alpha_{j}\\}_{j=0}^{\infty}\subset\mathbb{R}^{+}$ such that
for all $\displaystyle M\geq 0$,

$\mu_{1}(b\mathbf{F})=b^{\frac{1}{2}}\sum_{j=0}^{M}\alpha_{j}b^{\frac{-j}{4}}+\mathcal{O}\left(b^{\frac{1-M}{4}}\right)\,.$
(2.1)

Here,

$\displaystyle\alpha_{0}=\Lambda_{1}=\min_{x\in\Sigma}|\lambda^{\rm
Hess}_{2}(x)|^{1/2}\mathcal{X}_{1}\Big{(}\varepsilon(x)\Big{)}\,,$ (2.2)

$\displaystyle\varepsilon(x)$ and $\displaystyle\mathcal{X}_{1}$ are
introduced in (1.11) and (1.12) respectively.

Recall the definition of the left and right derivatives of
$\displaystyle\mu_{1}(b\mathbf{F})$,

$\displaystyle\frac{d}{db}\mu^{\pm}_{1}(b\mathbf{F})=\lim_{\varepsilon\to
0^{\pm}}\frac{\mu_{1}((b+\varepsilon)\mathbf{F})-\mu_{1}(b\mathbf{F})}{\varepsilon}\,,$

whose existence is guaranteed by the analytic perturbation theory [24], which
also gives for all $\displaystyle b>0$

$\frac{d}{db}\mu_{1}^{+}(b\mathbf{F})\leq\frac{d}{db}\mu_{1}^{-}(b\mathbf{F})\,.$
(2.3)

We introduce the new parameter $\displaystyle\beta=\sqrt{b}$ and the following
function

$\lambda(\beta)=\mu_{1}(\beta^{2}\mathbf{F})\,.$ (2.4)

Recall the following proposition [11, Page 27]

######  Proposition 2.2.

Let $\displaystyle g$ be a function such that for all
$\displaystyle\varepsilon\in(-1,\,1)$ we have

$|g(\beta+\varepsilon)-g(\beta)|\to 0\,,\qquad\text{as}\leavevmode\nobreak\
\beta\to+\infty\,.$ (2.5)

Suppose that there exists $\displaystyle\alpha\in\mathbb{R}$ such that

$\displaystyle\lambda(\beta)=\alpha\beta+g(\beta)+o(1)\,,\qquad\text{as}\leavevmode\nobreak\
\beta\to+\infty\,.$

Then the limits
$\displaystyle\lim_{\beta\to+\infty}\frac{d}{d\beta}\lambda(\beta)$ and
$\displaystyle\lim_{\beta\to+\infty}\frac{d}{d\beta}\lambda(\beta\mathbf{F})$
exist and

$\displaystyle\lim_{\beta\to+\infty}\frac{d}{d\beta}\lambda^{+}(\beta)=\lim_{\beta\to+\infty}\frac{d}{d\beta}\lambda^{-}(\beta)=\alpha\,.$

###### Proof of Theorem 1.1.

Choose $\displaystyle M=3$ in Theorem 2.1. The proof of Theorem 1.1 simply
follows by rewriting (2.1) in the following form (recall that
$\displaystyle\beta=\sqrt{b}$),

$\displaystyle\lambda(\beta)=\alpha_{0}\beta+g(\beta)+o(1)\,,$

where $\displaystyle
g(\beta)=\beta\sum_{j=1}^{3}\alpha_{j}\beta^{\frac{-j}{2}}$ and
$\displaystyle\lambda(\beta)$ is introduced in (2.4). For
$\displaystyle\varepsilon\in(-1,\,1)$ fixed, we have

$\displaystyle
g(\beta+\varepsilon)-g(\beta)\underset{\beta\to+\infty}{\sim}\frac{\alpha_{1}\varepsilon}{(\beta+\varepsilon)^{\frac{1}{2}}+\beta^{\frac{1}{2}}}$

which implies that (2.5) holds. We can use Proposition 2.2 in order to prove
the monotonicity of $\displaystyle\mu_{1}(b\mathbf{F})$. In fact,

$\displaystyle\lim_{\beta\to+\infty}\frac{d}{d\beta}\mu^{+}_{1}(\beta^{2}\mathbf{F})=\lim_{\beta\to+\infty}\frac{d}{d\beta}\mu^{-}_{1}(\beta^{2}\mathbf{F})=\alpha_{0}\,.$

To finish the proof, we write by the chain rule (recall that
$\displaystyle\beta=\sqrt{b}$),

$\displaystyle\lim_{b\to+\infty}b^{\frac{1}{2}}\frac{d}{db}\mu_{1}^{+}(b\mathbf{F})=\lim_{b\to+\infty}b^{\frac{1}{2}}\frac{d}{db}\mu_{1}^{-}(b\mathbf{F})=\frac{1}{2}\alpha_{0}\,.$

∎

##  3\. Breakdown of superconductivity

In this section we extend a result of Giorgi-Phillips [17] to the case where
the zero set of $\displaystyle B_{0}$ is self-intersecting. We will show that
the trivial solution $\displaystyle(0,\mathbf{F})$ is a global minimizer, when
$\displaystyle H/\kappa^{3}$ is large. We first recall a priori estimates for
a critical point $\displaystyle(\psi,\mathbf{A})$ of the G.L.functional (see
[2, Theorem 8.1]).

######  Theorem 3.1.

There exist positive constants $\displaystyle C_{1}$ and $\displaystyle C_{2}$
such that, if $\displaystyle\kappa>0$, $\displaystyle H>0$ and
$\displaystyle(\psi,\mathbf{A})$ is a solution of (1.10), then,

$\displaystyle\displaystyle\|(\nabla-i\kappa
H\mathbf{A})\psi\|_{L^{2}(\Omega)}$
$\displaystyle\displaystyle\leq\,\kappa\,\|\psi\|_{L^{2}(\Omega)}$ (3.1)
$\displaystyle\displaystyle\|(\mathbf{A}-\mathbf{F})\psi\|_{L^{2}(\Omega)}$
$\displaystyle\displaystyle\leq\frac{C_{1}}{H}\|\psi\|_{L^{4}(\Omega)}^{2}\|\psi\|_{L^{2}(\Omega)}$
(3.2) $\displaystyle\displaystyle\|(\nabla-i\kappa
H\mathbf{F})\psi\|_{L^{2}(\Omega)}$ $\displaystyle\displaystyle\leq
C_{2}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\,.$ (3.3)

In the next theorem, we prove that $\displaystyle(0,\mathbf{F})$ is the unique
minimizer of the functional when $\displaystyle H/\kappa^{3}$ is sufficiently
large.

######  Theorem 3.2.

Let $\displaystyle\Omega\subset\mathbb{R}^{2}$ be a smooth, bounded, simply-
connected open set. There exist positive constants $\displaystyle\bar{C}$ and
$\displaystyle\kappa_{0}$, such that, if

$\displaystyle H\geq\bar{C}\kappa^{3}\,,\qquad\kappa\geq\kappa_{0}\,,$

then $\displaystyle(0,\mathbf{F})$ is the unique solution to (1.10).

###### Proof.

We denote by $\displaystyle\mu_{1}^{N}(b\mathbf{F})$ the lowest eigenvalue of
the $\displaystyle\rm Schr\ddot{o}dinger$ operator $\displaystyle
P_{b\mathbf{F}}^{\Omega}$ with Neumann condition in $\displaystyle
L^{2}(\Omega)$. Assume that we have a non normal critical point
$\displaystyle(\psi,\mathbf{A})$ for
$\displaystyle\mathcal{E}_{\kappa,H,B_{0}}$. This means that
$\displaystyle(\psi,\mathbf{A})\in H^{1}(\Omega)\times H^{1}_{{\rm
div}}(\Omega)$ is a solution of (1.10) and

$\int_{\Omega}|\psi|^{2}\,dx>0\,.$

Using (3.1) with $\displaystyle b=\kappa H$, we get

$\displaystyle\frac{\|(\nabla-
ib\mathbf{F})\psi\|_{L^{2}(\Omega)}^{2}}{\|\psi\|^{2}_{L^{2}(\Omega)}}\leq\kappa^{2}\,,$

which implies by assumption that the lowest Neumann eigenvalue satisfies,

$\mu_{1}^{N}(b\mathbf{F})\leq\kappa^{2}\,.$

Using the leading order asymptotics in (2.1) with $\displaystyle b=\kappa H$
and $\displaystyle H\geq\bar{C}\kappa^{3}$, we obtain

$\Lambda_{1}=\lim_{\kappa H\longrightarrow+\infty}\frac{\mu_{1}^{N}(\kappa
H\mathbf{F})}{\sqrt{\kappa H}}\leq\bar{C}^{-\frac{1}{2}}\,.$ (3.4)

For $\displaystyle\bar{C}>(\Lambda_{1})^{-2}$, (3.4) yields a contradiction. ∎

Figure 4. Schematic representation of the levels of $\displaystyle H$ that
correspond to the trivial solution $\displaystyle(0,\mathbf{F})$.

##  4\. Criterion for the breakdown of superconductivity

The purpose of this section is to prove Theorem 1.3. One ingredient in the
proof is Proposition 4.1 below on the localization of the order parameter,
which is an adaptation of an analogous result obtained in [18] for the case
where the magnetic field has a non-empty zero set but not self-intersecting.
It says that, if $\displaystyle(\psi,\mathbf{A})$ is a critical point of the
functional in (1.8) and $\displaystyle H$ is of order
$\displaystyle\kappa^{3}$, then $\displaystyle|\psi|$ is concentrated near the
set $\displaystyle\Sigma=\\{x\in\Omega\leavevmode\nobreak\
:\leavevmode\nobreak\ B_{0}(x)=0,\&\,\nabla B_{0}(x)=0\\}$.

######  Proposition 4.1.

Let $\displaystyle\varepsilon>0$ and $\displaystyle\alpha\in\left(0,1\right)$.
There exist two positive constants $\displaystyle C$ and
$\displaystyle\kappa_{0}$ such that, if

$\kappa\geq\kappa_{0}\,,\quad H\geq\varepsilon\,\kappa^{3}\,,$ (4.1)

and $\displaystyle(\psi,\mathbf{A})$ is a solution of (1.10), then

$\|\psi\|^{2}_{L^{2}(\Omega)}\leq
C\,\kappa^{-\frac{\alpha}{2}}\|\psi\|^{2}_{L^{4}(\Omega)}\,.$ (4.2)

###### Proof.

Let $\displaystyle\ell=\kappa^{-\alpha}$ with $\displaystyle 0<\alpha<1$ and

$\displaystyle
D_{\ell}=\big{\\{}x\in\Omega:\operatorname{dist}(x,\partial\Omega)>\ell\leavevmode\nobreak\
\&\leavevmode\nobreak\ \operatorname{dist}(x,\Gamma)>\ell\big{\\}}\,.$

Let us introduce a cut off function $\displaystyle h\in
C^{\infty}_{c}(\Omega)$ satisfying

$\displaystyle 0\leq h\leq 1\leavevmode\nobreak\ {\rm in}\leavevmode\nobreak\
\Omega\,,\quad h=1\leavevmode\nobreak\ {\rm in}\leavevmode\nobreak\
D_{\ell}\,,\quad{\rm supp}\,h\subset D_{\ell/2}\quad{\rm and}\quad|\nabla
h|\leq\frac{C}{\ell}\quad{\rm in}\leavevmode\nobreak\ \Omega\,,$

where $\displaystyle C$ is a positive constant.
Adapting the proof by Helffer and Kachmar [18, Equation (6.6)], there exists a
positive constant $\displaystyle c$ such that,

$\displaystyle\kappa\,H\int_{\Omega}|B_{0}(x)|\,|h\psi|^{2}\,dx-c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|h\psi\|_{L^{4}(\Omega)}^{2}\leq\int_{\Omega}|(\nabla-i\kappa
H\mathbf{A})h\psi|^{2}\,dx\,.$

Using Cauchy’s inequality, we obtain

$\displaystyle
c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|h\psi\|_{L^{4}(\Omega)}^{2}\leq
c^{2}\|\psi\|_{L^{2}(\Omega)}^{2}+\kappa^{2}\|h\psi\|_{L^{4}(\Omega)}^{4}\,,$

which implies that

$\int_{\Omega}\,\left(\kappa\,H\,|B_{0}(x)|-\kappa^{2}\right)\,|h\psi|^{2}\,dx\leq\int_{\Omega}|(\nabla-i\kappa
H\mathbf{A})h\psi|^{2}\,dx-\kappa^{2}\|h\psi\|_{L^{2}(\Omega)}^{2}\\\
+c^{2}\|\psi\|_{L^{2}(\Omega)}^{2}+\kappa^{2}\|h\psi\|_{L^{4}(\Omega)}^{4}\,.$
(4.3)

Multiplying both sides of (1.10)a by $\displaystyle h^{2}\psi$, it results
from an integration by parts that,

$\int_{\Omega}\Big{(}|(\nabla-i\kappa
H\mathbf{A})h\psi|^{2}-\kappa^{2}|h\psi|^{2}\Big{)}\,dx=\int_{\Omega}|\nabla
h|^{2}|\psi|^{2}\,dx-\kappa^{2}\int_{\Omega}h^{2}|\psi|^{4}\,dx\,.$ (4.4)

Implementing (4.4) into (4.3), we get,

$\displaystyle\displaystyle\int_{\Omega}\,\left(\kappa\,H\,|B_{0}(x)|-\kappa^{2}\right)\,|h\psi|^{2}\,dx$
$\displaystyle\displaystyle\leq
c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla
h|^{2}|\psi|^{2}\,dx+\kappa^{2}\int_{\Omega}(h^{4}-h^{2})|\psi|^{4}\,dx$
$\displaystyle\displaystyle\leq
c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx\,.$

Here, we have used the fact that $\displaystyle h^{4}\leq h^{2}$ since
$\displaystyle 0\leq h\leq 1$.
As a consequence of the non degeneracy of $\displaystyle B_{0}$ outside
$\displaystyle\Gamma$, there exists a positive constant $\displaystyle
C(B_{0})$ such that

$|B_{0}(x)|\geq C(B_{0})\min\Big{(}{\rm dist}(x,\Sigma)^{2},{\rm
dist}(x,\Gamma)\Big{)}\geq C(B_{0})\,\ell^{2}\,,\qquad\forall x\in
D_{\ell}\,.$ (4.5)

Therefore, by using (4.1), we get,

$\displaystyle\Big{(}C(B_{0})\,\varepsilon\,\kappa^{4-2\alpha}-\kappa^{2}\Big{)}\int_{\Omega}|h\psi|^{2}\,dx\leq
c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx\,.$

Writing

$\displaystyle\int_{\Omega}|\psi|^{2}\,dx=\int_{\Omega}|h\psi|^{2}\,dx+\int_{\Omega}(1-h^{2})|\psi|^{2}\,dx$
(4.6)

and using the assumption on $\displaystyle h$, we have,

$\displaystyle\Big{(}C(B_{0})\,\varepsilon\,\kappa^{4-2\alpha}-\kappa^{2}-c^{2}\Big{)}\int_{\Omega}|h\psi|^{2}\,dx\leq\Big{(}c^{2}+C\,\kappa^{2\alpha}\Big{)}\int_{\Omega\setminus
D_{\ell}}|\psi|^{2}\,dx\,.$

For $\displaystyle\kappa$ large enough, $\displaystyle\displaystyle
C(B_{0})\,\varepsilon\,\kappa^{4-2\alpha}-\kappa^{2}-c^{2}\geq\frac{1}{2}C(B_{0})\,\varepsilon\,\kappa^{4-2\alpha}$
and

$\displaystyle\int_{\Omega}|h\psi|^{2}\,dx\leq
2\frac{C}{C(B_{0})\,\varepsilon}\,\kappa^{4(\alpha-1)}\int_{\Omega\setminus
D_{\ell}}|\psi|^{2}\,dx\,.$

Using the decomposition in (4.6) and that $\displaystyle 1-h^{2}$ vanishes
outside $\displaystyle D_{\ell}$, we get further,

$\displaystyle\int_{\Omega}|\psi(x)|^{2}\,dx\leq\left(2\frac{C}{C(B_{0})\,\varepsilon}\,\kappa^{4(\alpha-1)}+1\right)\int_{\Omega\setminus
D_{\ell}}|\psi|^{2}\,dx\leq 2\int_{\Omega\setminus D_{\ell}}|\psi|^{2}\,dx\,.$

Finally, by the Cauchy-Schwarz inequality,

$\displaystyle\int_{\Omega\setminus
D_{\ell}}|\psi|^{2}\,dx\leq|\Omega\setminus
D_{\ell}|^{1/2}\left(\int_{\Omega\setminus
D_{\ell}}|\psi|^{4}\,dx\right)^{\frac{1}{2}}\leq
C\,\kappa^{-\frac{\alpha}{2}}\left(\int_{\Omega}|\psi|^{4}\,dx\right)^{\frac{1}{2}}\,.$

This finishes the proof of the proposition. ∎

###### Proof of Theorem 1.3.

Suppose that $\displaystyle(\psi,\mathbf{A})$ is a solution of (1.10) with
$\displaystyle\|\psi\|_{L^{2}(\Omega)}\neq 0$. Then $\displaystyle
H\leq\bar{C}\kappa^{3}$, where $\displaystyle\bar{C}>0$ is the constant in
Theorem 3.2.

We will prove that $\displaystyle\mu_{1}(\kappa H\mathbf{F})<\kappa^{2}$. We
distinguish between three cases:

  * Case 1:

Let$\displaystyle H\geq\varepsilon\kappa^{3}$ and $\displaystyle
0<\varepsilon<\bar{C}$. Multiplying (1.10)a by $\displaystyle\psi$ in
$\displaystyle L^{2}(\Omega)$ and using (1.10)c, we observe that,

$\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}=-\int_{\Omega}\Big{(}\big{|}(\nabla-i\kappa
H\mathbf{A})\psi\big{|}^{2}-\kappa^{2}|\psi|^{2}\Big{)}\,dx\,.$ (4.7)

The Cauchy-Schwarz inequality yields for any
$\displaystyle\delta\in(0,\infty)$,

$\int_{\Omega}|(\nabla-i\kappa
H\mathbf{A})\psi|^{2}\,dx\geq(1-\delta)\int_{\Omega}|(\nabla-i\kappa
H\mathbf{F})\psi|^{2}\,dx+(1-\delta^{-1})\,(\kappa
H)^{2}\int_{\Omega}|(\mathbf{A}-\mathbf{F})\psi|^{2}\,dx\,.$ (4.8)

This implies that (after using the min-max principle),

$\displaystyle\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}\leq-\Big{(}\mu_{1}(\kappa
H\mathbf{F})+\kappa^{2}\Big{)}\,\|\psi\|_{L^{2}(\Omega)}^{2}+\delta\,\|(\nabla-i\kappa
H\mathbf{F})\psi\|^{2}_{L^{2}(\Omega)}+\delta^{-1}(\kappa
H)^{2}\|(\mathbf{A}-\mathbf{F})\psi\|_{L^{2}(\Omega)}^{2}\,.$

Using (3.2) and (3.3), we obtain,

$\displaystyle\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}\leq-\Big{(}\mu_{1}(\kappa
H\mathbf{F})+\kappa^{2}\Big{)}\,\|\psi\|_{L^{2}(\Omega)}^{2}+C\,\delta\,\kappa^{2}\,\|\psi\|_{L^{2}(\Omega)}^{2}+C^{\prime}\,\delta^{-1}\,\kappa^{2}\,\|\psi\|_{L^{2}(\Omega)}^{2}\|\psi\|_{L^{4}(\Omega)}^{4}\,.$

From (4.2), we find that,

$\displaystyle\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}\leq-\Big{(}\mu_{1}(\kappa
H\mathbf{F})+\kappa^{2}\Big{)}\,\|\psi\|_{L^{2}(\Omega)}^{2}+C\,\delta\,\kappa^{2-\frac{\alpha}{2}}\,\|\psi\|_{L^{4}(\Omega)}^{2}+C^{\prime}\,\delta^{-1}\,\kappa^{2-\frac{\alpha}{2}}\,\|\psi\|_{L^{4}(\Omega)}^{6}\,.$

Remembering that $\displaystyle\|\psi\|_{L^{2}(\Omega)}\neq 0$ and choosing
$\displaystyle\delta=\|\psi\|_{L^{4}(\Omega)}^{2}$, we get,

$\displaystyle\Big{(}\mu_{1}(\kappa
H\mathbf{F})+\kappa^{2}\Big{)}\,\|\psi\|_{L^{2}(\Omega)}^{2}\leq\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}\left(C\,\kappa^{-\frac{\alpha}{2}}-1\right)\,.$

We deduce for $\displaystyle\kappa$ sufficiently large
$\displaystyle\mu_{1}(\kappa H\mathbf{F})<\kappa^{2}$, which what we needed to
prove the first case.

  * Case 2:

Let $\displaystyle c_{0}\kappa^{-1}\leq H<\varepsilon\kappa^{3}$, where
$\displaystyle c_{0}>1$ is sufficiently large so that we can use Theorem 2.1
with $\displaystyle b=\kappa H$ and $\displaystyle M=0$. We observe as
$\displaystyle\kappa\to+\infty$,

$\displaystyle\displaystyle\mu_{1}(\kappa H\mathbf{F})$
$\displaystyle\displaystyle\leq(\kappa H)^{\frac{1}{2}}|\lambda^{\rm
Hess}_{2}(x)|^{1/2}\mathcal{X}_{1}\Big{(}\varepsilon(x)\Big{)}+C\,(\kappa
H)^{\frac{1}{4}}$
$\displaystyle\displaystyle<\kappa^{2}\epsilon^{\frac{1}{2}}|\lambda^{\rm
Hess}_{2}(x)|^{1/2}\mathcal{X}_{1}\Big{(}\varepsilon(x)\Big{)}+C\,\kappa\,.$

We can choose $\displaystyle\varepsilon>0$ such that
$\displaystyle\epsilon^{\frac{1}{2}}|\lambda^{\rm
Hess}_{2}(x)|^{1/2}\mathcal{X}_{1}\Big{(}\varepsilon(x)\Big{)}-1<0$ to get,

$\displaystyle\mu_{1}(\kappa H\mathbf{F})<\kappa^{2}\,,\qquad{\rm
as}\leavevmode\nobreak\ \kappa\longrightarrow+\infty\,.$

  * Case 3:

$\displaystyle H<c_{0}\kappa^{-1}$. Let $\displaystyle x_{0}\in\Omega$ and let
$\displaystyle\chi_{\ell}\in C_{c}^{\infty}(\Omega)$ be a cut-off function
satisfying:

$0\leq\chi_{\ell}\leq 1\leavevmode\nobreak\ {\rm in}\leavevmode\nobreak\
\Omega\,,\quad{\rm supp}\,\chi_{\ell}\subset
B(x_{0},\ell)\,,\quad\chi_{\ell}=1\leavevmode\nobreak\ {\rm
in}\leavevmode\nobreak\ B\left(x_{0},\frac{\ell}{2}\right)\quad{\rm
and}\quad|\nabla\chi_{\ell}|\leq\frac{C}{\ell}\,.$ (4.9)

We may write,

$\displaystyle\mu_{1}(\kappa
H\mathbf{F})\leq\frac{\displaystyle\int_{\Omega}|(\nabla-i\kappa
H\mathbf{F})\chi_{\ell}|^{2}\,dx}{\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}}\,.$

Using the assumptions on $\displaystyle\chi_{\ell}$ and the fact that
$\displaystyle\mathbf{F}\in C^{\infty}(\overline{\Omega})$, a trivial estimate
is,

$\displaystyle\displaystyle\int_{\Omega}|(\nabla-i\kappa
H\mathbf{F})\chi_{\ell}|^{2}\,dx$
$\displaystyle\displaystyle\leq\int_{B(x_{0},\ell)}|\nabla\chi_{\ell}|^{2}\,dx+\kappa^{2}H^{2}\int_{B(x_{0},\ell)}|\mathbf{F}\chi_{\ell}|^{2}\,dx$
$\displaystyle\displaystyle\leq C\,(1+(\kappa\,H\,\ell)^{2})\,.$ (4.10)

This implies that

$\displaystyle\Big{(}\mu_{1}(\kappa
H\mathbf{F})-\kappa^{2}\Big{)}\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq
C\Big{(}1+(\kappa\,H\,\ell)^{2}\Big{)}-\,(\kappa\ell)^{2}\,.$

We select $\displaystyle\ell=\kappa^{-\frac{1}{2}}$ and remember that
$\displaystyle H<c_{0}\,\kappa^{-1}$. We find that,

$\displaystyle\Big{(}\mu_{1}(\kappa
H\mathbf{F})-\kappa^{2}\Big{)}\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}<C\,(1+\kappa^{-1})-\kappa\,,$

and consequently, for $\displaystyle\kappa$ sufficiently large,

$\displaystyle\mu_{1}(\kappa H\mathbf{F})<\kappa^{2}\,.$

∎

## References

  * [1] K. Attar. The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field, Ann. I. H. Poincaré, vol. 32, 325-345 (2015).
  * [2] K. Attar. Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model, Nonlinear Analysis, vol. 139, 1-54 (2016).
  * [3] W. Assaad, A. Kachmar, M.P. Sundqvist. The Distribution of Superconductivity Near a Magnetic Barrier, Commun. Math. Phys. Vol. 366 (1), 269-332 (2019).
  * [4] W. Assaad. The breakdown of superconductivity in the presence of magnetic steps, Com. Cont. Mat.. Vol. 23(2), 2050005 (2021).
  * [5] V. Bonnaillie-Noël, S. Fournais. Superconductivity in domains with corners, Reviews in Mathematical Physics, Vol. 19(6), 607-637 (2007).
  * [6] V. Bonnaillie-Noël, N. Raymond. Breaking a magnetic zero locus: model operators and numerical approach, ZAMM Z. Angew. Math. Mech. Vol. 95, 120-139 (2015).
  * [7] M. Correggi, N. Rougerie. On the Ginzburg-Landau Functional in the Surface Superconductivity Regime, Commun. Math. Phys. vol. 332, 1297-1343 (2014).
  * [8] M. Dauge, J.P. Miqueu, N. Raymond. On the semiclassical Laplacian with magnetic field having self-intersecting zero set, J. Spectr. Theory. Vol 10(4), 1211-1252 (2020).
  * [9] N. Dombrowski, N. Raymond. Semiclassical analysis with vanishing magnetic fields, J. Spectr. Theory. vol 3, 423-464 (2013).
  * [10] S. Fournais, B. Helffer. On the third critical field in Ginzburg-Landau theory, Comm. Math. Phys. Vol. 226 (1), 153-196, (2006).
  * [11] S. Fournais, B. Helffer. Strong diamagnetism for general domains and application, Ann. Inst. Fourier. Vol 57(7), 2389-2400 (2007).
  * [12] S. Fournais, B. Helffer. On the Ginzburg-Landau critical field in three dimensions, Commun. Pure Appl. Math. vol 62(2), 215-241 (2009).
  * [13] S. Fournais, B. Helffer. Spectral Methods in Surface Superconductivity, Progress Nonlinear Differential Equations Appl., vol. 77, Birkh$\displaystyle\ddot{a}$user, Boston, (2010).
  * [14] S. Fournais, A. Kachmar. On the transition to the normal phase for superconductors surrounded by normal conductors, Journal of Differential Equations, vol. 247, 1637-1672 (2009).
  * [15] S. Fournais, A. Kachmar. Nucleation of bulk superconductivity close to critical magnetic field, Advances in Mathematics, vol. 226, 1213-1258 (2011).
  * [16] S. Fournais, M. Persson Sundqvist. Lack of Diamagnetism and the Little-Parks Effect, Commun. Math. Phy., 337, 191-224 (2015).
  * [17] T. Giorgi, D. Phillips. The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, Siam J. Math. Anal. vol. 30, 341-359 (1999).
  * [18] B. Helffer, A. Kachmar. The Ginzburg-Landau functional with vanishing magnetic field, Arch. Rational Mech. Anal., vol. 218, 55-122 (2015).
  * [19] B. Helffer, A. Kachmar. Thin domain limit and counterexamples to strong diamagnetism, Reviews in Mathematical Physics, vol. 33 (2), 2150003 (2021).
  * [20] A. Kachmar, X. B. Pan. Superconductivity and the Aharonov-Bohm effect, C. R. Acad. Sci. Paris, Ser. I. vol.  357, 216-220 (2019).
  * [21] A. Kachmar, X. B. Pan. Oscillatory patterns in the Ginzburg-Landau model driven by the Aharonov-Bohm potential, J. Funct. Anal. vol. 279 (10), 108718 (2020).
  * [22] A. Kachmar, M. P. Sundqvist. Counterexample to strong diamagnetism for the magnetic Robin Laplacian, Math. Phys. Anal. Geom. vol. 23 (27), 23-27 (2020).
  * [23] A. Kachmar, M. Wehbe. Averaging of magnetic fields and applications, arXiv:2003.04415 .
  * [24] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, (1976).
  * [25] K. Lu, X-B.Pan. Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Physica D., vol. 127, 73-104 (1999).
  * [26] W.A. Little, R.D. Parks. Observation of quantum periodicity in the transition temperature of a superconducting cylinder, Phys. Rev. Lett. vol.9(1), 9-12 (1962)
  * [27] R. Montgomery. Hearing the zero locus of a magnetic field, Comm. Math. Phys., vol. 168, 651-675 (1995).
  * [28] X.B. Pan, K.H. Kwek. Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains, Trans. Am. Math. Soc. vol. 354 (10), 4201-4227 (2002).
  * [29] X.B. Pan. Surface superconductivity in applied magnetic fields above $\displaystyle H_{C_{2}}$, Comm. Math. Phys. vol. 228, 327-370 (2002).
  * [30] N. Raymond. Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2, Ann. I. H. Poincaré. vol. 10, 95-122 (2009).
  * [31] E. Sandier, S.Serfaty. Ginzburg-Landau minimizers near the first critical field have bounded vorticity, Calc. Var. Partial Differential Equations. vol. 17 (1), 17-28 (2003).
  * [32] E. Sandier, S.Serfaty. The Decrease of Bulk-Superconductivity Close to the Second Critical Field in the Ginzburg-Landau Model, SIAM J. Math. Anal.. vol. 34 (4), 939-956 (2003).