RUP-24-2

YITP-24-11

Convexity restoration from hairy black hole in Einstein-Maxwell-charged scalar
system in AdS

Takaaki Ishii and Yu Nakayama

Department of Physics, Rikkyo University, Nishi-Ikebukuro, Toshima-ku, Tokyo
171-8501, Japan

Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa
Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

###### Abstract

In the Einstein-Maxwell-charged scalar system with a negative cosmological
constant in arbitrary dimensions higher than three, there exists a horizonless
charged soliton solution, which we construct explicitly for an arbitrary mass
of the scalar in perturbative series in small charge. We find that the
stability of the soliton is determined by the validity of the AdS weak gravity
conjecture. The existence of a stable soliton might endanger the convexity of
the (free) energy as a function of the charge because the phase transition
between the soliton and the extremal Reissner-Nordstrom black hole would be
discontinuous. We, however, argue that the existence of the hairy black hole
solution circumvents the violation of convexity. The thermodynamic properties
of the hairy black hole show that the phase transition becomes continuous
irrespective of whether the AdS weak gravity conjecture holds. When it holds,
the phase transition occurs between the soliton and the hairy black hole, and
when it is violated, the phase transition occurs between the extremal
Reissner-Nordstrom black hole and the hairy black hole.

##  1 Introduction

Convexity of the (free) energy is at the heart of thermodynamic stability [1].
Still, it is a non-trivial problem whether this is realized in statistical
mechanics with a given microscopic Hamiltonian. While it is generically very
difficult to verify the convexity, the gravitational system may admit explicit
analysis because we can read the thermodynamic properties from the classical
solutions, circumventing microscopic statistical mechanical computations. For
example, a Schwarzschild black hole in asymptotically flat spacetime or a
small Schwarzschild-AdS black hole shows negative specific heat, which means
that the convexity (as a function of temperature) is violated while the
microscopic understanding of the violation as well as its implication would be
quite challenging.

In this paper, we would like to focus on the convexity of the (lowest) energy
as a function of the conserved charge. Again, in generic statistical
mechanics, while we expect the convexity of the (free) energy as a function of
charge from the thermodynamic stability, it is a highly non-trivial task to
verify it in interacting systems. The gravitational benchmark would be an
extremal Reissner-Nordstrom black hole, which does show convexity.

The recent developments in theoretical physics, however, seem to suggest that
the (extremal) Reissner-Nordstrom black hole is not the lowest energy state
with a given charge. The weak gravity conjecture [2], for example, states that
it should be unstable in any consistent theory of quantum gravity (see e.g.
[3] for review). Indeed, with (light) charged matter, we observe its
instability caused by various mechanisms such as classical instability from
superradiance or quantum tunneling (e.g. Schwinger effect). Then the task of
verifying the convexity requires a study of the onset of such instability and
becomes more involved, which we will undertake in this paper.

In terms of the AdS/CFT correspondence, the convexity of the global energy in
the AdS space-time as a function of the charge is related to the charge
convexity conjecture of the conformal dimensions in conformal field theories.
This was first proposed in [4] and then checked in many examples
[5][6][7][8][9].111The convexity and the superadditivity are closely related
but they can differ. Strictly speaking, in the original paper [4], the charge
superadditivity is conjectured. See Appendix A for the connection. See [5] as
well. A counter-example in three-dimensional conformal field theories was
found in [10], whose significance should be understood better. A closely
related subject is the interpretation of the weak gravity conjecture in terms
of conformal field theories [11]. We do have a general picture in many
examples, but the precise bound has not been established.

In this paper, we investigate solutions of the Einstein-Maxwell-charged scalar
system with an arbitrary mass for the scalar in asymptotically AdS spacetime.
We find there exists a horizonless charged soliton solution, which may have a
lower global energy than the extremal Reissner-Nordstrom black hole. We
constructed them in perturbative series in small charge, and they are
generalizations of massless cases in five dimensions [12][13][14] to arbitrary
dimensions with arbitrary mass.222A direct motivation of our paper comes from
Figures 2 and 3 in [14], which show the apparent violation of the charge
convexity. Their figures were “schematic” as they say, and if we closely
follow their computations, their results do not violate the convexity. Compare
them with Fig. 1. See also charged solitons in four dimensions with and
without scalar masses [15][16][17]. We first point out that the existence of
such a charged soliton might endanger the convexity of the lowest energy as a
function of the charge because the phase transition between the soliton and
the extremal Reissner-Nordstrom black hole would be discontinuous.333In this
paper, we consider the canonical ensemble at zero temperature. Meanwhile, the
phase transition between the (finite temperature) charged AdS black hole and
soliton is often discussed in the grand canonical ensemble [18] through the
grand potential. The convexity of the (canonical) free energy discussed in
this paper is equivalent to the concavity of the grand potential due to the
thermodynamic relation [1]. We, however, argue that the convexity is restored
thanks to the existence of the hairy black hole solution. In particular, we
show that the existence of the hairy black hole makes the phase transition
continuous.

When the AdS weak gravity conjecture is satisfied by the scalar field, the
soliton is lighter than the extremal Reissner-Nordstrom black hole of the same
charge in the small charge limit. If we increase the charge, the charged
soliton becomes a hairy black hole, and the latter will be the lowest energy
state. When the AdS weak gravity conjecture is violated, the soliton is
heavier than the extremal Reissner-Nordstrom black hole in the small charge
limit. If we increase the charge, the extremal Reissner-Nordstrom black hole
becomes unstable to form the hairy black hole, which will be the lowest energy
state. In both cases, the phase transition will be continuous and the
convexity will be restored. A schematic plot for the case when the AdS weak
gravity conjecture is satisfied is given in Fig. 1.

Figure 1: A schematic plot for the mass (or global energy) of the (extremal)
Reissner-Nordstrom black hole (blue), charged soliton (orange), and the hairy
black hole (red) in $d=3$. Here, the AdS weak gravity conjecture is satisfied.
See section 4 for details.

The organization of the paper is as follows. In section 2, we construct the
charged soliton solution in perturbative series in small charge and compare
the thermodynamic properties with those of the extremal Reissner-Nordstrom
black hole. In section 3, we study the thermodynamic properties of a hairy
black hole. In section 4, we study the phase transition between these
solutions to see if the convexity is violated or not. Section 5 is devoted to
discussion. Appendices contain details of calculations.

##  2 Reissner-Nordstrom black hole and charged soliton

We study solutions of the Einstein-Maxwell-charged scalar theory described by
the action

$\displaystyle S=\int d^{d+1}x\sqrt{-g}\left(\frac{1}{16\pi
G_{N}}(R-2\Lambda)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-|D_{\mu}\phi|^{2}-m_{\phi}^{2}|\phi|^{2}\right)\
,$ (1)

where $\Lambda=-d(d-1)/2$ is the negative cosmological constant in units in
which the AdS radius is unity, and the gauge covariant derivative is defined
as $D_{\mu}\phi=\partial_{\mu}\phi-ieA_{\mu}\phi$ with $e$ being the $U(1)$
coupling constant. For convenience, we will parameterize the mass of the
charged scalar by the conformal dimension $\Delta$ as
$m_{\phi}^{2}=\Delta(\Delta-d)$. In this paper, we assume $d\geq 3$.

Without the scalar hair $(\phi=0)$, the supposedly lowest energy solution with
charge $Q$ is the Reissner-Nordstrom-AdS black hole solution. In the
conventional radial coordinate, the Reissner-Nordstrom-AdS black hole has the
metric and gauge field:

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{d-1}\ ,$ (2)
$\displaystyle A_{\mu}dx^{\mu}$
$\displaystyle=\left(\mu-\frac{q}{r^{d-2}}\right)dt\ ,$ (3) $\displaystyle
f(r)$
$\displaystyle=r^{2}+1-\frac{m}{r^{d-2}}+\frac{d-2}{d-1}\frac{\kappa^{2}q^{2}}{r^{2(d-2)}}\
,$ (4)

where we use the notation $\kappa^{2}=8\pi G_{N}$. The event horizon of the
black hole is located at $r=r_{h}$ given by the largest root of $f(r_{h})=0$.
We will fix the $U(1)$ gauge by demanding the gauge field satisfies
$A_{t}(r_{h})=0$, which specifies that $\mu$ is the chemical potential of the
$U(1)$. From these conditions, we obtain

$\displaystyle m$
$\displaystyle=r_{h}^{d-2}\left(r_{h}^{2}+1+\frac{d-2}{d-1}\frac{\kappa^{2}q^{2}}{r_{h}^{2(d-2)}}\right)\
,$ (5) $\displaystyle\mu$ $\displaystyle=\frac{q}{r_{h}^{d-2}}\ .$ (6)

The thermodynamic properties of the Reissner-Nordstrom-AdS black hole can be
obtained from the above solution in a standard manner. The global energy and
charge are related to the parameters in the solution as

$\displaystyle E$ $\displaystyle=\frac{d-1}{16\pi G_{N}}\omega_{d-1}m\ ,$ (7)
$\displaystyle Q$ $\displaystyle=(d-2)\omega_{d-1}q\ ,$ (8)

where $\omega_{d-1}$ denotes the volume of a unit $(d-1)$-sphere,
$\omega_{d-1}=2\pi^{\frac{d}{2}}/\Gamma(\frac{d}{2})$. The chemical potential
can then be written as a function of the charge as

$\displaystyle\mu=\frac{Q}{(d-2)r_{h}^{d-2}\omega_{d-1}}\ .$ (9)

The temperature and the entropy are given by

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi}f^{\prime}(r_{h})\ ,$ (10)
$\displaystyle S$ $\displaystyle=\frac{r_{h}^{d-1}\omega_{d-1}}{4G_{N}}\ .$
(11)

These thermodynamic quantities satisfy the first law of thermodynamics
$dE=TdS+\mu dQ$.

In the following, we focus on the zero temperature limit, where the Reissner-
Nordstrom black hole becomes extremal. Note that in the extremal limit, we
have $\mu=\frac{\partial E}{\partial Q}$ because the extremal black hole has
$T=0$ so $F=E-TS=E$, and $\mu=\frac{\partial F}{\partial Q}=\frac{\partial
E}{\partial Q}$. Then, we want to express the energy $E$ as a function of
charge $Q$ explicitly. We can obtain the closed formulae valid for any charge
in some particular dimensions, say $d=3,4$ and $5$.444Eq. (64) is a quadratic,
cubic, or quartic equation of $r_{h}^{2}$ in $d=3,4$, or $5$, respectively,
and can be solved for $r_{h}^{2}$, but it cannot be algebraically solved in
general in $d\geq 6$, where the equation is quintic or of higher degrees. For
example, in $d=3$, we find (see e.g. [19][20])

$\displaystyle
E_{d=3}=\frac{4\pi\sqrt{6}}{9\kappa^{2}}\sqrt{-1+\sqrt{1+6\kappa^{2}q^{2}}+6\kappa^{2}q^{2}\left(3+\sqrt{1+6\kappa^{2}q^{2}}\right)}\
,$ (12)

where $q$ is related to $Q$ as in (8). In this paper, instead, we will only
use the small charge expansion. The explicit form of the energy and chemical
potential in the small charge expansion is given in general dimensions by (see
Appendix B for calculations)

$\displaystyle E$
$\displaystyle=\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}Q\left(1+\frac{1}{2}\widetilde{Q}^{\frac{2}{d-2}}-\frac{d^{2}}{8(d-2)^{2}}\widetilde{Q}^{\frac{4}{d-2}}+\frac{d^{4}}{16(d-2)^{4}}\widetilde{Q}^{\frac{6}{d-2}}+\cdots\right),$
(13) $\displaystyle\mu$
$\displaystyle=\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}\left(1+\frac{d}{2(d-2)}\widetilde{Q}^{\frac{2}{d-2}}-\frac{d^{2}(d+2)}{8(d-2)^{3}}\widetilde{Q}^{\frac{4}{d-2}}+\frac{d^{4}(d+4)}{16(d-2)^{5}}\widetilde{Q}^{\frac{6}{d-2}}+\cdots\right),$
(14)

where

$\displaystyle\widetilde{Q}\equiv\frac{\kappa
Q}{\omega_{d-1}\sqrt{(d-1)(d-2)}}=\kappa q\sqrt{\frac{d-2}{d-1}}\ .$ (15)

At this point, we can verify that the global energy of the extremal Reissner-
Nordstrom black hole is a convex function of the charge. Since $E(0)=0$, this
implies that it is superadditive (see Appendix A). It is a non-trivial
prediction of the black hole solution that the expansion parameter is
$Q^{\frac{2}{d-2}}$, which is dimension specific (rather than $Q$), and it
should be contrasted with the charged soliton solution whose expansion
parameter is $Q$ in any dimensions as we will see later.

The slope of the energy as a function of the charge sets the weak gravity
conjecture bound. In this paper, we say that the weak gravity conjecture is
satisfied when there is a state whose energy-charge ratio is smaller than that
of the extremal Reissner-Nordstrom black hole in the small charge limit:

$\displaystyle\frac{E(Q)}{Q}\leq\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}\ .$
(16)

Alternatively, we say that the weak gravity conjecture is violated when there
is no state whose energy-charge ratio is smaller than that of the extremal
Reissner-Nordstrom black hole in the small charge limit: all the states
satisfy

$\displaystyle\frac{E(Q)}{Q}\geq\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}\ .$
(17)

In the literature, there are some different notions of AdS weak gravity
conjecture than that used here (see e.g. [11][21][22][23][24][25] in relation
to CFTs). At least for the (non-interacting) charged scalar matter, we will
see that this bound is relevant in our analysis of the hairy black hole. We
also note that the bound we consdier is the same as the bound used in [26] to
avoid the formulation of the naked singularity in the Einstein-Maxwell-charged
scalar system in AdS (in $d=3$ studied there). We will discuss this point
further below when we introduce the critical coupling $e_{c}$.

In addition to the extremal Reissner-Nordstrom black hole, in this theory,
there exists a charged soliton solution, that has non-zero scalar condensate.
When the weak gravity conjecture holds, we expect that the soliton solution,
at least in the small charge limit, has a lower energy than the extremal
Reissner-Nordstrom black hole.

We have explicitly constructed the horizonless soliton solution within a
perturbative expansion with respect to the scalar condensate (or equivalently
charge). See Appendix C for details. They are reliable in the small charge
limit, which we will focus on. Some examples of the solutions in various
dimensions with various scalar masses can be found in Appendix C. Here we
present the final result of the energy and the chemical potential of the
soliton as functions of the charge up to the second order in small charge
expansion:

$\displaystyle E$
$\displaystyle=(d-2)\omega_{d-1}\left[\frac{\Delta}{e}q+\frac{\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}{2\,\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}\left(1-\frac{(d-2)\Delta^{2}(\Delta-d/2)}{(d-1)(\Delta-d/4)}\frac{\kappa^{2}}{e^{2}}\right)q^{2}\right]$
(18)
$\displaystyle=Q\left[\frac{\Delta}{e}+\frac{\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}{2\,\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}\left(1-\frac{(d-2)\Delta^{2}(\Delta-d/2)}{(d-1)(\Delta-d/4)}\frac{\kappa^{2}}{e^{2}}\right)q\right]\
,$ (19) $\displaystyle\mu$
$\displaystyle=\frac{\Delta}{e}+\frac{\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}{\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}\left(1-\frac{(d-2)\Delta^{2}(\Delta-d/2)}{(d-1)(\Delta-d/4)}\frac{\kappa^{2}}{e^{2}}\right)q\
,$ (20)

where $q$ is related to $Q$ as in (8). We see that $\mu=\frac{\partial
E}{\partial Q}$, which implies $TS=0$ because $E=F$. There is no horizon, so
the gravitational entropy is zero. In the Euclidean continuation of the
horizonless soliton, the periodicity of the compactified time direction is
arbitrary, so is the temperature. In this paper, we consider zero temperature.

Let us introduce the notion of the critical coupling $e_{c}$. When the charge
$Q$ is small, the energy of the extremal Reissner-Nordstrom black hole (14)
and that of the soliton (20) behave as

$\displaystyle E_{RN}\simeq\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}Q\ ,\qquad
E_{sol}\simeq\frac{\Delta}{e}Q\ .$ (21)

We identify the critical coupling $e_{c}$ as the value of the coupling that
equates these energies,

$\displaystyle e_{c}=\Delta\kappa\sqrt{\frac{d-2}{d-1}}\ .$ (22)

This critical coupling is exactly the lower bound of the gauge coupling
constant to preserve cosmic censorship discussed in [26] for $d=3$.555In [26],
the lower bound of the scalar field charge (i.e. $U(1)$ coupling constant)
$q_{W}$ is given in their normalization as $q_{W}^{(\text{theirs})}=\Delta$,
where the AdS radius is set unity, but our normalization of the fields and
coupling constant is different from theirs. Rescaling the gauge field and
gauge coupling constant in [26] so that they match ours, the lower bound reads
$q_{W}^{(\text{ours})}=\Delta\kappa/\sqrt{2}$, which is nothing but (22) for
$d=3$: $q_{W}^{(\text{ours})}=e_{c}|_{d=3}$. We expect (22) gives the
generalization of the bound of [26] to general dimensions $d\geq 3$.

The critical coupling connects the soliton with the weak gravity conjecture
mentioned above. Given $\Delta$ and small $Q$, the small charged soliton has a
smaller energy than the extremal Reissner-Nordstrom black hole when $e>e_{c}$.
This is equivalent to saying that the weak gravity conjecture is satisfied by
the charged soliton. We can also say that gravity is weaker than the electric
force and a horizonless charged soliton can be formed without gravitationally
collapsing to a black hole. Alternatively, the small charged soliton has a
larger energy than the extremal Reissner-Nordstrom black hole when $e<e_{c}$.
This is equivalent to saying that the weak gravity conjecture does not hold
because the electric force is weaker than gravity.

Another point we would like to discuss about the properties of the charged
soliton is the convexity of the energy. The energy of the soliton is convex
$\frac{d^{2}E}{dQ^{2}}>0$ when the coupling constant satisfies

$\displaystyle
e^{2}>\frac{(d-2)\Delta^{2}(\Delta-d/2)}{(d-1)(\Delta-d/4)}\kappa^{2}=\frac{\Delta-d/2}{\Delta-d/4}e_{c}^{2}\
.$ (23)

The right-hand side is not positive for $\Delta\leq d/2$ in $d\geq 4$ and
$3/4<\Delta\leq 3/2$ in $d=3$.666Recall that the range that $\Delta$ can take
is $\Delta\geq d/2-1$ (unitarity bound). This means that, when $\Delta$ is in
these ranges, the energy of the soliton is convex for any coupling constant
$e$. Otherwise, the right-hand side is positive. Accordingly, if the coupling
constant is too small, the energy of the soliton is not convex. However, such
a coupling is smaller than the critical coupling (22) (for $\Delta>d/2$) below
which the soliton is not the configuration with the smallest energy, and the
convexity is irrelevant. If we further believe in the weak gravity conjecture,
this parameter region does not arise.

We here observe that in $d=3,4$ the inequality (23) obtained from solving
Einstein’s equation is equivalent to demanding positivity of the binding
energy $\gamma_{\text{grav}}+\gamma_{\text{photon}}>0$ from the perturbative
formula in [27] ($d=4$) and [28] ($d=3$). See also eq. (2.5) of [4] and
discussions there. We expect a similar comparison can be done in other
dimensions.

The only subtle case is $1/2<\Delta<3/4$ for $d=3$, where the right-hand side
of the bound (23) is larger than the critical coupling, i.e.
$e^{2}>4e_{c}^{2}$. This means that the convexity can be violated near $e\sim
e_{c}$. Our formula implies that the attractive force from the graviton
exchange (i.e. $\gamma_{\text{grav}}$ in [28]) becomes infinite at
$\Delta=3/4$ and continues to be large down to the unitarity bound
$\Delta=1/2$. We might want to say that some extended notion of weak gravity
conjecture is violated in this case even though $e>e_{c}$. If the soliton is
stable, which we do not know, it may violate the charge convexity. We,
however, note that our formula for $1/2<\Delta<3/4$ is obtained from the
analytic continuation of $\Delta>3/4$ and we have not constructed an explicit
solution in this range, which may be singular. To avoid the subtlety, we
assume $\Delta>3/4$ for $d=3$ hereafter.

##  3 Hairy black hole

As we have seen, if the weak gravity conjecture is violated, even with the
existence of the soliton solution, the extremal Reissner-Nordstrom black hole
solution has a lower energy in the small charge limit. What happens if we
increase the charge? Does it show a phase transition to the charged soliton?
Or, does it continue to be the lowest energy state? In the following, we will
argue that at a certain critical charge, the extremal Reissner-Nordstrom black
hole forms a scalar hair. The formation of the hair is due to superradiant
instability of the Reissner-Nordstrom-AdS black hole [17][29][30][31].
Similarly, if the weak gravity conjecture holds, the lowest energy solution
with a given charge is the soliton (in the small charge limit), but if we
increase the charge further (as well as the energy), it collapses into a hairy
black hole. In this section, we will predict the thermodynamic properties of
the hairy black hole in the Einstein-Maxwell-charged scalar system, and in the
next section, we will study the nature of the phase transition.

We have already seen that both the horizonless charged soliton solution and
the Reissner-Nordstrom black hole solution are compatible with the
thermodynamic relation $\mu=\frac{\partial E}{\partial Q}$ at zero
temperature. It is natural to expect that the hairy black hole solution also
obeys the thermodynamic relations. This observation enables us to use the
thermodynamic equilibrium argument to predict the property of the hairy black
hole as well. As discussed in [14], when the charge is small, we will regard
the extremal hairy black hole as a thermodynamic mixture of the extremal
Reissner-Nordstrom black hole and the horizonless charged soliton. We express
the charge and the energy of the hairy black hole as a non-interacting sum as

$\displaystyle Q_{hairy}$ $\displaystyle=aQ_{RN}+bQ_{sol}\ ,$ (24)
$\displaystyle E_{hairy}$ $\displaystyle=aE_{RN}(Q_{RN})+bE_{sol}(Q_{sol})\ ,$
(25)

where the mixing parameters $a,b\geq 0$ are determined by the thermodynamic
equilibrium condition

$\displaystyle\mu_{RN}(Q_{RN})=\mu_{sol}(Q_{sol})\ .$ (26)

The thermodynamic equilibrium condition also demands that the temperature must
be equal, and our assumption that the charged soliton has zero temperature
makes it equilibrium with the extremal Reissner-Nordstrom black hole.

This simple picture gives a determination of the critical charge $Q_{c}$ above
which we will see that the hairy black hole becomes the lowest energy state.
When $e<e_{c}$ (i.e. weak gravity conjecture is violated), it is determined by
equating the chemical potential of the Reissner-Nordstrom black hole at
$Q_{RN}=Q_{c}$ and that of the soliton at $Q_{sol}=0$:
$\mu_{RN}(Q_{c})=\mu_{sol}(0)$. In the same way, when $e>e_{c}$ (i.e. weak
gravity conjecture holds), it is determined by equating the chemical potential
of the soliton at $Q_{sol}=Q_{c}$ and that of the Reissner-Nordstrom black
hole at $Q_{RN}=0$: $\mu_{RN}(0)=\mu_{sol}(Q_{c})$.

The critical charge $Q_{c}$ can be analytically expressed in arbitrary
dimensions.777For $d=3$, we assume $\Delta>\frac{3}{4}$. See discussions after
(23). For $e<e_{c}$, let us take $e^{2}=e_{c}^{2}(1-\theta)<e_{c}^{2}$, where
$\theta>0$ is a small parameter. With this coupling, the chemical potential of
the soliton at $Q_{sol}=0$ is given by

$\displaystyle\mu_{sol}(0)=\frac{\Delta}{e}=\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}\left(1+\frac{\theta}{2}+\cdots\right)\
.$ (27)

Comparing this with $\mu_{RN}(Q_{c})$ in (14) to the subleading terms, we
obtain

$\displaystyle
Q_{c}=\frac{\omega_{d-1}\sqrt{(d-1)(d-2)^{d-1}}}{\kappa}\left(\frac{\theta}{d}\right)^{\frac{d-2}{2}}\
.$ (28)

For $e>e_{c}$, we take $e^{2}=e_{c}^{2}(1+\theta)>e_{c}^{2}$, where
$\theta>0$. The charge of the soliton that balances with this shift of the
coupling can be parametrized as $Q_{c}=q_{sol}=b_{c}\theta$. Then, the series
expansion of (20) in $\theta$ becomes

$\displaystyle\mu_{sol}(Q_{c})=\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}+\left(\frac{b_{c}d\,\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}{(4\Delta-d)\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}-\frac{1}{2\kappa}\sqrt{\frac{d-1}{d-2}}\right)\theta+\cdots\
.$ (29)

The leading order term is the same as $\mu_{RN}(0)$ from eq. (14). Then, from
$\mu_{RN}(0)=\mu_{sol}(Q_{c})$, the critical charge is given by the vanishing
of the subleading term in (29) as

$\displaystyle Q_{c}=\frac{4\Delta-d}{2\kappa
d}\omega_{d-1}\sqrt{(d-1)(d-2)}\frac{\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}{\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}\theta\
.$ (30)

Now we have determined the critical charge, let us study the energy of the
hairy black hole above $Q_{c}$. We solve the equilibrium condition (25) with
suitable ansatz in the small charge limit. The results are summarized below
for $d=3,4,5$ and $Q>Q_{c}$. We will also give comments on the hairy black
holes in $d\geq 6$. For $Q<Q_{c}$, we find that the energy of the hairy black
hole is larger than that of the soliton or extremal Reissner-Nordstrom black
hole (and one of the mixing parameters in (25) becomes negative), so below we
will not discuss the hairy black holes with $Q<Q_{c}$. For derivation, see
Appendix D.

###  3.1 Hairy black hole in $e<e_{c}$

When the coupling constant is smaller than the critical value as
$e^{2}=e_{c}^{2}(1-\theta)$, we can construct a hairy black hole with
$Q\sim\theta^{(d-2)/2}$.

##### AdS4 $(d=3)$

The critical coupling is

$\displaystyle Q_{c}=\omega_{2}q_{c}\ ,\qquad
q_{c}=\frac{\sqrt{6}}{3\kappa}\theta^{1/2}\ .$ (31)

The energy of the hairy black hole in $Q>Q_{c}$ is

$\displaystyle E=\frac{\omega_{2}}{\kappa^{2}}m_{hairy}\ ,\qquad
m_{hairy}=m_{RN}-\frac{3\times
2^{2\Delta-5}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\kappa^{4}(q^{2}-q_{c}^{2})^{2}\
,$ (32)

where

$\displaystyle m_{RN}=\sqrt{2}\kappa
q+\frac{\kappa^{3}q^{3}}{2\sqrt{2}}+O(q^{5})\ .$ (33)

The energy satisfies $m_{hairy}<m_{RN}$ if $\Delta>3/4$. The energy of the
hairy black hole is a convex function of the charge as we can see at
$O(q^{3})$ of the Reissner-Nordstrom black hole (33).

##### AdS5 $(d=4)$

The critical coupling is

$\displaystyle Q_{c}=2\omega_{3}q_{c}\ ,\qquad
q_{c}=\frac{\sqrt{6}}{4\kappa}\theta\ .$ (34)

The energy of the hairy black hole in $Q>Q_{c}$ is

$\displaystyle E=\frac{3\omega_{3}}{2\kappa^{2}}m_{hairy}\ ,\qquad
m_{hairy}=m_{RN}-\frac{2(2\Delta-1)}{3(3\Delta-2)}\kappa^{2}(q-q_{c})^{2}\ ,$
(35)

where

$\displaystyle m_{RN}=\frac{2\sqrt{6}}{3}\kappa q+\frac{2}{3}\kappa^{2}q^{2}\
.$ (36)

For $\Delta>1$ ($=\frac{d}{2}-1$ with $d=4$), we have
$\frac{2}{3}<\frac{2\Delta-1}{3\Delta-2}<1$. Hence, we find that the energy of
the hairy black hole satisfies $m_{hairy}<m_{RN}$ and also is a convex
function of the charge.

##### AdS6 $(d=5)$

The critical coupling is

$\displaystyle Q_{c}=3\omega_{4}q_{c}\ ,\qquad
q_{c}=\frac{6}{5\sqrt{5}\kappa}\theta^{3/2}\ .$ (37)

The energy of the hairy black hole in $Q>Q_{c}$ is

$\displaystyle E=\frac{2\omega_{4}}{\kappa^{2}}m_{hairy}\ ,\qquad m_{hairy}$
$\displaystyle=\sqrt{3}\kappa q+\left(\frac{\sqrt{3}}{2}\kappa
q\theta-\frac{6\sqrt{15}}{125}\theta^{5/2}\right)$ (38)
$\displaystyle=m_{RN}-\left(\frac{3^{5/6}\kappa^{5/3}q^{5/3}}{2^{5/3}}-\frac{\sqrt{3}}{2}\kappa
q\theta+\frac{6\sqrt{15}}{125}\theta^{5/2}\right)\ ,$ (39)

where

$\displaystyle m_{RN}=\sqrt{3}\kappa
q+\frac{3^{5/6}\kappa^{5/3}q^{5/3}}{2^{5/3}}\ .$ (40)

One can check that $m_{hairy}<m_{RN}$ and
$\frac{dm_{hairy}}{dq}|_{q=q_{c}}=\frac{dm_{RN}}{dq}|_{q=q_{c}}$ by expanding
(39) around $q=q_{c}$, where $|q-q_{c}|\ll\theta^{3/2}$, as

$\displaystyle
m_{hairy}=m_{RN}-\frac{5\sqrt{15}}{36\sqrt{\theta}}\kappa^{2}(q-q_{c})^{2}+O\left((q-q_{c})^{3}\right)\
.$ (41)

The result (39) given up to $O(q)$ is not sufficient to discuss the convexity
of $m_{hairy}$, so we also calculate the next order $O(q^{2})$ term and find
the convexity as

$\displaystyle\frac{d^{2}m_{hairy}}{dq^{2}}=\frac{5\kappa^{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}>0\
.$ (42)

###  3.2 Hairy black hole in $e>e_{c}$

When the coupling constant is larger than the critical value as
$e^{2}=e_{c}^{2}(1+\theta)$, we can construct a hairy black hole with
$Q\sim\theta$.

##### AdS4 $(d=3)$

The critical coupling is

$\displaystyle Q_{c}=\omega_{2}q_{c}\ ,\qquad
q_{c}=\frac{2^{2\Delta-3/2}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{3\kappa\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\theta\
.$ (43)

The energy of the hairy black hole in $Q>Q_{c}$ is

$\displaystyle E=\frac{\omega_{2}}{\kappa^{2}}m_{hairy}\ ,\qquad m_{hairy}$
$\displaystyle=\sqrt{2}\kappa
q-\frac{2^{2\Delta-3}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{3\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\theta^{2}$
(44)
$\displaystyle=m_{sol}-\frac{3\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}\kappa^{2}(q-q_{c})^{2}\
,$ (45)

where

$\displaystyle m_{sol}=\sqrt{2}\kappa
q+\left(\frac{3\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}\kappa^{2}q^{2}-\frac{\kappa
q\theta}{\sqrt{2}}\right)\ .$ (46)

The energy satisfies $m_{hairy}<m_{sol}$ if $\Delta>3/4$. Because the
coefficient of $q^{2}$ in $m_{hairy}$ vanishes, we need to go to the next
order to check the convexity. From the calculations including $O(\theta^{3})$
terms, we find

$\displaystyle\frac{d^{2}m_{hairy}}{dq^{2}}=\frac{3\sqrt{2}}{2}\kappa^{3}(q-q_{c})\
,$ (47)

so the convexity is satisfied in $Q>Q_{c}$.

##### AdS5 $(d=4)$

The critical coupling is

$\displaystyle Q_{c}=2\omega_{3}q_{c}\ ,\qquad
q_{c}=\frac{\sqrt{6}(2\Delta-1)}{4(\Delta-1)\kappa}\theta\ .$ (48)

The energy of the hairy black hole in $Q>Q_{c}$ is

$\displaystyle E=\frac{3\omega_{3}}{2\kappa^{2}}m_{hairy}\ ,\qquad
m_{hairy}=m_{sol}-\frac{2(\Delta-1)^{2}}{3(2\Delta-1)(3\Delta-2)}\kappa^{2}(q-q_{c})^{2}\
,$ (49)

where

$\displaystyle m_{sol}=\frac{2\sqrt{6}}{3}\kappa
q+\left(\frac{2(\Delta-1)}{3(2\Delta-1)}\kappa^{2}q^{2}-\frac{\sqrt{6}}{3}\kappa
q\theta\right)\ .$ (50)

The energy satisfies $m_{hairy}<m_{sol}$. Because
$1-\frac{\Delta-1}{3\Delta-2}$ is positive in $\Delta>1$, the convexity is
satisfied.

##### AdS6 $(d=5)$

The critical coupling is

$\displaystyle Q_{c}=3\omega_{4}q_{c}\ ,\qquad
q_{c}=\frac{2^{2\Delta-2}\sqrt{3}\,\Gamma(\Delta-3/2)^{2}\,\Gamma(\Delta+1/2)}{5\kappa\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}\theta\
.$ (51)

The energy of the hairy black hole in $Q>Q_{c}$ is

$\displaystyle E=\frac{2\omega_{4}}{\kappa^{2}}m_{hairy}\ ,\qquad
m_{hairy}=m_{sol}-\frac{3^{1/4}\,\Gamma(\Delta)^{5/2}\,\Gamma(2\Delta-5/2)^{5/2}}{2^{5\Delta-6}\,\Gamma(\Delta-3/2)^{5}\,\Gamma(\Delta+1/2)^{5/2}}\kappa^{5/2}(q-q_{c})^{5/2}\
,$ (52)

where

$\displaystyle m_{sol}=\sqrt{3}\kappa
q+\left(\frac{5\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{2^{2\Delta}\,\Gamma(\Delta-3/2)^{2}\,\Gamma(\Delta+1/2)}\kappa^{2}q^{2}-\frac{\sqrt{3}}{2}\kappa
q\theta\right)\ .$ (53)

We can see that $m_{hairy}<m_{sol}$. The convexity of $m_{hairy}$ is satisfied
because the energy of the soliton (53) is already convex and it dominates over
the added term in (52).

Let us conclude this section with comments on hairy black holes in $d\geq 6$.
When $e<e_{c}$ (i.e. the weak gravity conjecture is violated), the similar
construction outlined in Appendix D leads to the expression
$m_{hairy}(q)=m_{RN}(q)+C_{RN}(q-q_{c})^{2}+O((q-q_{c})^{3})$ around
$q=q_{c}$. The coefficient $C_{RN}$ is such that $q^{d/d-2}$ term in the small
$q$ expansion of $m_{hairy}(q)$ vanishes and $m_{hairy}(q)$ is an integer
power series in $q$. The above form of $m_{hairy}(q)$ implies that the phase
transition is continuous between the Reissner-Nordstrom black hole and the
hairy black hole (i.e. the first derivatives of $m$ with respect to $q$ are
the same at $q=q_{c}$). The convexity of $m_{hairy}(q)$ can be checked from
the explicit computation of the $O(q^{2})$ term.

When $e>e_{c}$ (i.e. the weak gravity conjecture holds), the similar ansatz in
Appendix D instead leads to the expression
$m_{hairy}(q)=m_{sol}(q)+C_{sol}(q-q_{c})^{\frac{d-2}{2}}$ around $q=q_{c}$.
To determine the precise numerical constant $C_{sol}$ (independent of
$\theta$), we need the explicit form of the soliton to higher orders than we
calculated. This is beyond the scope of this paper and we do not attempt.
Nevertheless, we can still argue that the energy of the hairy black hole must
be convex for $d\geq 6$ because $q^{2}$ term in $m_{sol}(q)$ is dominant over
$C(q-q_{c})^{\frac{d-2}{2}}$ in the small charge expansions of $m_{hairy}(q)$
and we know its convexity (as discussed at the end of the previous section).

##  4 Phase transition and restoration of convexity

Let us now study the would-be phase transition between the Reissner-Nordstrom
solution and the soliton solution that we constructed in section 2. Later in
this section, within the parameter space where our solution is valid, we will
show that this type of phase transition never happens due to the existence of
the hairy black hole solution. We, nevertheless, first ask ourselves what
would happen if the hairy black hole solution did not exist.

First, consider the case when the weak gravity conjecture is violated (i.e.
$e<e_{c}$). In the small charge limit, the extremal Reissner-Nordstrom black
hole has the lowest energy. Depending on the parameter, it may or may not
happen that if we increase the charge, the charged soliton solution starts to
possess a lower energy than the extremal Reissner-Nordstrom black hole. When
it does, generically there is a (would-be) discontinuous phase transition from
the Reissner-Nordstrom black hole to the charged soliton solution. It is
(generically) discontinuous because the first derivative of the energy with
respect to the charge (i.e. chemical potential) at the phase transition point
is discontinuous.

Similarly, when the weak gravity conjecture holds (i.e. $e>e_{c}$), in the
small charge limit the charged soliton has the lowest energy. Then there may
exist a phase transition to the extremal Reissner-Nordstrom black hole if we
increase the charge. If it does, again the phase transition would be
discontinuous (see Fig. 1).

One can compute the would-be phase transition point more explicitly in the
small charge limit where our solutions are reliable. In AdS4, solving
$m_{RN}(q_{*})=m_{sol}(q_{*})$ in the small charge limit (i.e. $|\theta|\ll
1$), we obtain for $e^{2}=e_{c}^{2}(1+\theta)>e_{c}^{2}$

$\displaystyle
q_{*}=\frac{2^{2\Delta-1/2}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{3\kappa\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\theta\
,$ (54)

while there is no $q_{*}>0$ for $e<e_{c}$ (we assume $\Delta>3/4$), so the
(would-be) phase transition occurs when $e>e_{c}$ (i.e. the AdS weak gravity
conjecture holds).

In AdS5, a similar calculation gives for $e^{2}=e_{c}^{2}(1-\theta)<e_{c}^{2}$

$\displaystyle q_{*}=\frac{\sqrt{6}(2\Delta-1)}{2\kappa\Delta}\theta\ ,$ (55)

while there is no $q_{*}>0$ for $e>e_{c}$, so the (would-be) phase transition
occurs when $e<e_{c}$ (i.e. the AdS weak gravity conjecture does not hold).

In AdSd+1 with $d\geq 5$, we observe that the (would-be) phase transition
occurs when $e<e_{c}$ (i.e. the AdS weak gravity conjecture does not hold).
For $e^{2}=e_{c}^{2}(1-\theta)<e_{c}^{2}$, we obtain

$\displaystyle
q_{*}=\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}\,\theta^{\frac{d-2}{2}}\ .$ (56)

while there is no $q_{*}>0$ for $e>e_{c}$. Note that in all these cases, the
scaling of $q_{*}$ with respect to $\theta$ is the same as that for $q_{c}$
studied in the previous section.

When the phase transition is discontinuous, the convexity of the energy is
violated. The second derivative of the energy as a function of the charge
becomes negative (in the delta function sense) at the phase transition, and we
can easily find three charges around the phase transition point where the
convexity inequality $\lambda E(Q_{1})+(1-\lambda)E(Q_{2})\geq E(\lambda
Q_{1}+(1-\lambda)Q_{2})$ is violated.

Note, however, that this does not mean that the discontinuous phase transition
causes the violation of the superadditivity. The convex function with $E(0)=0$
is always superadditive (see Appendix A), but the converse is not necessarily
true. Indeed, we can show $E(Q)=\mathrm{min}(E_{RN}(Q),E_{sol}(Q))$ is
superadditive. The elementary proof can be found in Appendix A.

The existence of the hairy black hole solution that we found in the previous
section completely changes the story above. This is because it is always the
case that the hairy black hole enters the phase diagram before the would-be
phase transition discussed above (i.e. $q_{c}<q_{*}$). See Fig. 1 for a
schematic plot of this situation. In particular, the would-be violation of the
convexity of the energy is circumvented as we will see.

The study in section 3 shows that above $Q_{c}$, the hairy black hole is the
lowest energy solution of the system with a given charge $Q$.888Logically
speaking, we do not exclude the possibility that a non-spherical solution,
which we have not studied, has lower energy than the spherical hairy black
hole. The salient feature of the thermodynamic property of the hairy black
hole solution is that not only the energy but also the chemical potential is a
continuous function with respect to $Q$ (while the higher derivative can be
discontinuous). This can be explicitly seen from our formula in the previous
section where the energy difference is $O((q-q_{c})^{2})$ or higher. It is
continuous but is not analytic because the higher derivative shows the
discontinuity at $Q=Q_{c}$. We further realize that it is a convex function
(at least within the small charge limit we have studied). We therefore
conclude that the convexity of the energy is restored by the existence of the
hairy black hole solution. Since the convex function with $E(0)=0$ is
superadditive (see Appendix A), the energy is superadditive with respect to
charge.

Let us briefly discuss the nature of the phase transition from the AdS/CFT
viewpoint. The big difference between the (extremal) Reissner-Nordstrom black
hole and the hairy black hole is that the former does not have scalar hair.
The scalar hair or the scalar condensation induced in the hairy black hole
suggests that in the dual CFT, the scalar operator which is dual to $\phi$
(with conformal dimension $\Delta$) has a non-zero expectation value on the
cylinder in this heavy state. It is distinct from the one-particle state that
is dual to low-dimensional single trace operators with zero scalar expectation
value. In the planer black hole (or black membrane), the analogous phase
transition discussed here is referred to as a “holographic superconductor”
[32][33] (or holographic superfluid), but we should point out that the phase
transition here is about the properties of operators on the plane (or states
in the cylinder).999Schematically, one may say that while the Reissner-
Nordstrom black hole is dual to $\mathrm{Tr}(\Phi^{m})$ with $m\sim O(N^{2})$,
the hairy black hole obtained as adding soliton should look like
$\mathrm{Tr}(\Phi^{m})(\mathrm{Tr}(\Phi^{2}))^{l}$. It may be interesting to
study the implication of the phase transition in terms of the recent studies
of black hole operators in CFTs [34][35].

##  5 Discussions

In this paper, we have studied the convexity of the lowest energy of the
Einstein-Maxwell-charged scalar system in AdS under the presence of a hairy
black hole. In the small charge limit, the existence of the hairy black hole
solution restores the would-be violated convexity due to the phase transition
between a charged soliton and a Reissner-Nordstrom black hole. It should be
interesting to perform (numerical) analysis in the larger charge regime and
examine if the convexity remains true.

Indeed, there has been some interest in studying the lowest conformal
dimensions of conformal field theories as a function of the charge, in
particular in the large charge expansion. Recent studies show that the large
charge behavior of the generic conformal field theories should be $E(Q)\sim
Q^{\frac{d}{d-2}}$ [36][37]. We can easily check that the extremal Reissner-
Nordstrom black hole satisfies this scaling. It was also numerically verified
in charged soliton (with $\Delta=2$ in $d=3$) in [38]. It will be interesting
to verify the behavior in the large charge hairy black holes with arbitrary
$\Delta$. Note that in some supergravity, they found BPS hairy black hole
solutions where $E(Q)\propto Q$ rather than $Q^{\frac{d}{d-2}}$ scaling.
Apparently, the presence of the moduli allows the linear behavior. It is
generally believed that it should not exhibit such a behavior without
supersymmetry, but there is no proof.101010By introducing $|\phi|^{4}$
interaction with a suitable coefficient, we expect that we can adjust the
$Q^{2}$ term in $E(Q)$ in the charged soliton solutions so that it has a
linear behavior in the small charge limit. In relation, it is highly doubtful
but is not proved if there is any solution whose scaling behavior is in
between $Q^{\alpha}$ where $1<\alpha<\frac{d}{d-2}$. It is an interesting
study if we can or cannot engineer such solutions in gravity.

The study of the large charge behavior should shed light on the analytic
property of large charge expansions. From the effective field theory
viewpoint, it was not obvious if the large charge expansion is converging
(down to $Q\sim 0$), has a finite radius of convergence, or is just an
asymptotic expansion. The extreme Reissner-Nordstrom solution tells that the
function is smooth for the entire range of charge, but the large charge
expansion has a finite radius of convergence.111111One can see that there is a
branch cut in the negative $Q$ in $E_{RN}(Q)$, so one can read the radius of
convergence in the positive $Q\sim\frac{1}{\kappa}$, which can be verified
against d’Alembert’s ratio test. In this paper, we have shown that the hairy
black hole solution shows a continuous phase transition, at which the function
is not analytic. Whether the large charge expansion breaks down much above
this phase transition point should be studied further in detail.

In this paper, we have studied charged solitons and hairy black holes under
the Dirichlet boundary condition at the AdS boundary. It is an interesting
question to allow more general boundary conditions such as the Robin boundary
condition to study the charged solitons and hairy black holes [16][39]. In the
dual field theory perspective, it would correspond to double trace
deformations [40][41]. The stability condition may change and whether the
convexity remains to hold is a non-trivial question to be addressed.

Our analysis of the charged solitons and hairy black holes was limited to
$d\geq 3$ in this paper. For $d=2$, it is necessary to consider scalar hair
around the charged BTZ black hole, and (extremal) hairy black holes can be
drastically different from higher dimensions. The charge convexity conjecture
in [4] is also restricted to $d>2$, and it remains an open question if similar
arguments can be established for $d=2$ CFT. It would be interesting if this
situation could be approached from three-dimensional gravity duals.

Finally, it may be an interesting question to address the higher derivative
gravitational corrections to the small charge (hairy) black holes and
solitons. To satisfy the weak gravity conjecture in the Minkowski space-time,
it is often suggested [42][43] that the higher derivative corrections make the
extremal Reissner-Nordstrom black hole lighter. If the same mechanism is at
work in the AdS, it may conflict with the convexity conjecture. The
competition then may lead to new constraints on effective gravitational field
theory in AdS.

## Acknowledgments

The authors would like to thank Yoshihiko Abe, Nick Dorey, and Toshifumi Noumi
for fruitful discussions. T.I. is supported in part by JSPS KAKENHI Grant
Number 19K03871. Y.N. is in part supported by JSPS KAKENHI Grant Number
21K03581.

The authors thank the Yukawa Institute for Theoretical Physics at Kyoto
University, where this work was initiated during the YITP workshop “Strings
and Fields 2023”.

##  Appendix A Convexity vs Superadditivity

In this appendix, we collect some mathematical facts about convex functions in
relation to superadditivity.

When a function $f(x)$ is convex between $x_{1}\leq x\leq x_{2}$, for all
$0\leq\lambda\leq 1$, $f(x)$ satisfies the inequality

$\displaystyle\lambda f(x_{1})+(1-\lambda)f(x_{2})\geq f(\lambda
x_{1}+(1-\lambda)x_{2})\ .$ (57)

When $f(x)$ is twice differentiable, $f^{\prime\prime}(x)\geq 0$ in a segment
implies the convexity in the same segment. The proof is based on Taylor’s
theorem

$\displaystyle
f(x)=f(x_{0})+f^{\prime}(x_{0})(x-x_{0})+\frac{f^{\prime\prime}(x_{*})}{2}(x-x_{0})^{2}\
,$ (58)

where $x_{0}\leq x_{*}\leq x$ . By noting $f^{\prime\prime}(x_{*})\geq 0$ and
setting $x_{0}=\lambda x_{1}+(1-\lambda)x_{2}$ with $x=x_{1}$ and $x=x_{2}$,
we have

$\displaystyle f(x_{1})$ $\displaystyle\geq
f(x_{0})+f^{\prime}(x_{0})(1-\lambda)(x_{1}-x_{2})\ ,$ (59) $\displaystyle
f(x_{2})$ $\displaystyle\geq f(x_{0})+f^{\prime}(x_{0})\lambda(x_{2}-x_{1})\
.$ (60)

Adding $\lambda$ times the first line and $(1-\lambda)$ times the second line,
we obtain (57).

When $f(x)$ is convex in $x\geq 0$ and $f(0)=0$, $f(x)$ is superadditive (i.e.
$f(a)+f(b)\leq f(a+b)$). To show this, we first set $x_{2}=0$ in (57) to
obtain $f(\lambda x_{1})\leq\lambda f(x_{1})$. Then, noting
$f(a)=f(\frac{a}{a+b}(a+b))\leq\frac{a}{a+b}f(a+b)$, we obtain

$\displaystyle f(a)+f(b)\leq\frac{a}{a+b}f(a+b)+\frac{b}{a+b}f(a+b)=f(a+b)\ .$
(61)

Let $g_{1}(x)$ and $g_{2}(x)$ be two twice differentiable convex functions
with $g_{1}(0)=g_{2}(0)=0$, $g_{1}(x)\leq g_{2}(x)$ when $0\leq x\leq x_{*}$,
and $g_{2}(x)\leq g_{1}(x)$ when $x_{*}\leq x$. Let
$f(x)=\mathrm{min}[g_{1}(x),g_{2}(x)]$ for $x\geq 0$. Generically, $f(x)$ is
not convex, but we will show that it is superadditive.

When $a,b\geq x_{*}$ or $a+b\leq x_{*}$, the convexity of $g_{1}(x)$ or
$g_{2}(x)$ immediately implies the claim. The nontrivial case is when $a,b\leq
x_{*}$ but $a+b\geq x_{*}$, or when $a\geq x_{*},b\leq x_{*}$. In the former
case,

$\displaystyle f(a+b)-f(a)-f(b)=g_{2}(a+b)-g_{1}(a)-g_{1}(b)\geq
g_{2}(a+b)-g_{2}(a)-g_{2}(b)\geq 0\ ,$ (62)

and in the latter case,

$\displaystyle f(a+b)-f(a)-f(b)=g_{2}(a+b)-g_{2}(a)-g_{1}(b)\geq
g_{2}(a+b)-g_{2}(a)-g_{2}(b)\geq 0\ ,$ (63)

so the claim holds in all cases.

##  Appendix B Small charge extremal Reissner-Nordstrom-AdS black hole

We want to express the energy and chemical potential (6) of the extremal
Reissner-Nordstrom black hole as functions of the charge. At $T=0$, from
$f^{\prime}(r_{h})=0$, the horizon radius and charge are related as

$\displaystyle
q=\frac{r_{h}^{d-2}}{(d-2)\kappa}\sqrt{(d-1)(d-2+d\,r_{h}^{2})}\ .$ (64)

When the charge $q$ is small, this equation can be inverted by a series
expansion as

$\displaystyle
r_{h}^{2}=\widetilde{Q}^{\frac{2}{d-2}}-\frac{d}{(d-2)^{2}}\widetilde{Q}^{\frac{4}{d-2}}+\frac{d^{2}(d+1)}{2(d-2)^{4}}\widetilde{Q}^{\frac{6}{d-2}}-\frac{d^{4}(d+2)}{3(d-2)^{6}}\widetilde{Q}^{\frac{8}{d-2}}+\cdots\
,$ (65)

where $\widetilde{Q}$ is defined in (15). Substituting the above expansion
into $E$ and $\mu$ in (6) and (8), we obtain (14).

##  Appendix C Explicit solution for the charged soliton

In this appendix, we present the explicit form of the solution of the
equations of motion describing a horizonless charged soliton in the small
charge expansion. We begin with the spherically symmetric ansatz for the
static solution of the Einstein-Maxwell-charged scalar system,

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+g(r)dr^{2}+r^{2}d\Omega_{d-1}^{2}\ ,$ (66)
$\displaystyle A_{\mu}dx^{\mu}$ $\displaystyle=A_{t}(r)dt\ ,$ (67)
$\displaystyle\phi$ $\displaystyle=\phi(r)\ ,$ (68)

where $d\Omega_{d-1}^{2}$ is the line element of a unit $S^{d-1}$. We have
chosen the gauge so that the complex scalar field $\phi$ is a real function of
$r$. Substituting the ansatz into the equations of motion of (1), we obtain a
set of equations to be solved:

$\displaystyle\frac{d-1}{2}rf^{\prime}+\left(\frac{(d-1)(d-2)}{2}(1-g)+\Lambda
r^{2}g-\kappa^{2}r^{2}\phi^{\prime
2}+\kappa^{2}m_{\phi}^{2}r^{2}g\phi^{2}\right)f$ (69)
$\displaystyle+\frac{\kappa^{2}}{2}r^{2}A_{t}^{\prime
2}-\kappa^{2}e^{2}r^{2}gA_{t}^{2}\phi^{2}$ $\displaystyle=0\ ,$ (70)
$\displaystyle\frac{d-1}{2}rg^{\prime}-\left(\frac{(d-1)(d-2)}{2}(1-g)+\Lambda
r^{2}g+\kappa^{2}r^{2}\phi^{\prime
2}+\kappa^{2}m_{\phi}^{2}r^{2}g\phi^{2}\right)g$ (71)
$\displaystyle-\frac{\kappa^{2}g}{2f}r^{2}A_{t}^{\prime
2}-\frac{\kappa^{2}e^{2}r^{2}g^{2}}{f}A_{t}^{2}\phi^{2}$ $\displaystyle=0\ ,$
(72) $\displaystyle
A_{t}^{\prime\prime}+\left(\frac{d-1}{r}-\frac{f^{\prime}}{2f}-\frac{g^{\prime}}{2g}\right)A_{t}^{\prime}-2e^{2}g\phi^{2}A_{t}$
$\displaystyle=0\ ,$ (73)
$\displaystyle\phi^{\prime\prime}+\left(\frac{d-1}{r}+\frac{f^{\prime}}{2f}-\frac{g^{\prime}}{2g}\right)\phi^{\prime}+\left(\frac{e^{2}A_{t}^{2}}{f}-m_{\phi}^{2}\right)g\phi$
$\displaystyle=0\ .$ (74)

Here, ′ denotes the derivative with respect to $r$.

Around the background with $\phi=0$, we will solve these equations in
perturbative series with respect to a small parameter $\epsilon$ as

$\displaystyle f(r)$
$\displaystyle=f_{(0)}(r)+f_{(2)}(r)\epsilon^{2}+f_{(4)}(r)\epsilon^{4}+\cdots\
,$ (75) $\displaystyle g(r)$
$\displaystyle=g_{(0)}(r)+g_{(2)}(r)\epsilon^{2}+g_{(4)}(r)\epsilon^{4}+\cdots\
,$ (76) $\displaystyle A_{t}(r)$
$\displaystyle=A_{t(0)}(r)+A_{t(2)}(r)\epsilon^{2}+A_{t(4)}(r)\epsilon^{4}+\cdots\
,$ (77) $\displaystyle\phi(r)$
$\displaystyle=\phi_{(1)}(r)\epsilon+\phi_{(3)}(r)\epsilon^{3}+\phi_{(5)}(r)\epsilon^{5}+\cdots\
.$ (78)

The strategy is to first solve the scalar equation at $O(\epsilon)$, where the
integration constants are fixed by the asymptotic behavior. Then, at
$O(\epsilon^{2})$, we first solve $g$ and $f$. Finally, we solve $A_{t}$. One
integration constant appearing in $A_{t}$ is not fixed at this point but will
be fixed by studying the regularity of the scalar function in
$O(\epsilon^{3})$. In this sense, the equations at $O(\epsilon^{2})$ and
$O(\epsilon^{3})$ are bundled. Then, we continue to $O(\epsilon^{4})$ for $f$,
$g$, and $A_{t}$, and so on.

In the zeroth order in $\epsilon$, the background is the empty AdS spacetime
with a constant gauge field specified by the chemical potential $\mu_{(0)}$,

$\displaystyle f_{(0)}(r)=1+r^{2}\ ,\qquad g_{(0)}(r)=\frac{1}{1+r^{2}}\
,\qquad A_{t(0)}(r)=\mu_{(0)}\ ,$ (79)

where we assume $\mu_{(0)}>0$. The empty AdS spacetime has zero charge.

At $O(\epsilon)$, we turn on the scalar field. The equation for $\phi_{(1)}$
reads

$\displaystyle\phi_{(1)}^{\prime\prime}+\left(\frac{d-1}{r}+\frac{2r}{1+r^{2}}\right)\phi_{(1)}^{\prime}+\frac{1}{1+r^{2}}\left(\frac{e^{2}\mu_{(0)}^{2}}{1+r^{2}}-\Delta(\Delta-d)\right)\phi_{(1)}$
$\displaystyle=0\ ,$ (80)

where we used $m_{\phi}^{2}=\Delta(\Delta-d)$. At the center of the AdS $r=0$,
this second-order differential equation has regular and diverging solutions.
To solve this equation (without horizon), we impose regularity at $r=0$. In
the AdS boundary $r\to\infty$, we impose the absence of the source discussed
as follows.

In $r\to\infty$, the behavior of the scalar field (not specific to
$O(\epsilon)$ but to any orders) is
$\phi=c_{0}(r^{\Delta-d}+\cdots)+c_{1}(r^{-\Delta}+\cdots)$, where $c_{0}$ and
$c_{1}$ are constants. For the absence of the source in the boundary field
theory, we require $c_{0}=0$. When $\Delta>d/2$, $r^{-\Delta}$ decays faster
than $r^{\Delta-d}$. Hence, setting $c_{0}=0$ corresponds to keeping the
subleading behavior of $\phi$. This case is called the standard quantization.
When $\Delta<d/2$, $r^{-\Delta}$ decays slower than $r^{\Delta-d}$, but the
leading behavior is normalizable for $\Delta$ in the range $d/2-1<\Delta<d/2$
[44]. This case is called the alternative quantization. The border
$\Delta=d/2$ is the case that the scalar mass saturates the Breitenlohner-
Freedman (BF) bound [45][46], where we have $\phi=c_{0}(r^{-d/2}\log
r+\cdots)+c_{1}(r^{-d/2}+\cdots)$.

In the absence of the scalar field source $c_{0}=0$, the equation (80) is
solved at discrete values of $\mu_{(0)}$ satisfying

$\displaystyle\mu_{(0)}=\frac{\Delta+2n}{e}\ ,$ (81)

where $n$ is a non-negative integer. This is nothing but the normal mode
frequency $\omega$ of the massive scalar field in the global AdS as
$\omega=e\mu_{(0)}=\Delta+2n$. At this $\mu_{(0)}$, the regular solution to
(80) is given by the hypergeometric function as

$\displaystyle\phi_{(1)}=(1+r^{2})^{\frac{\Delta}{2}+n}{}_{2}F_{1}\left(\frac{d}{2}+n,\Delta+n;\frac{d}{2};-r^{2}\right)\
,$ (82)

whose normalization is absorbed by $\epsilon$. Using this as the seed, we
proceed to the higher orders in the perturbative series. In the following, we
consider the solution with the lowest energy for a given $\Delta$, and hence
we set $n=0$. For $n=0$, the above chemical potential and scalar field
solution become

$\displaystyle\mu_{(0)}=\frac{\Delta}{e}\
,\qquad\phi_{(1)}=\frac{1}{(1+r^{2})^{\Delta/2}}\ .$ (83)

For $O(\epsilon^{2})$ and higher, we have solved these equations for integer
values of $\Delta$ in various dimensions $d$. Some sample solutions are
presented here.

For $d=4,\,\Delta=4\ (m_{\phi}^{2}=0)$, we reproduce the results in [12]. We
obtain

$\displaystyle f(r)$
$\displaystyle=1+r^{2}-\frac{8(3+3r^{2}+r^{4})}{9(1+r^{2})^{3}}\kappa^{2}\epsilon^{2}+\cdots\
,$ (84) $\displaystyle g(r)$
$\displaystyle=\frac{1}{1+r^{2}}+\frac{8r^{2}(3+r^{2})}{9(1+r^{2})^{5}}\kappa^{2}\epsilon^{2}+\cdots\
,$ (85) $\displaystyle A_{t}(r)$
$\displaystyle=\frac{4}{e}+\left(\frac{3e}{14}-\frac{(3+3r^{2}+r^{4})e}{6(1+r^{2})^{3}}-\frac{32\kappa^{2}}{21e}\right)\epsilon^{2}+\cdots\
,$ (86) $\displaystyle\phi(r)$
$\displaystyle=\frac{\epsilon}{(1+r^{2})^{2}}+\cdots\ .$ (87)

Here, we present the results up to $O(\epsilon^{2})$, but the perturbative
solutions can be obtained also in higher orders in $\epsilon$. Once the
equations of motion are solved, the mass $m$, chemical potential $\mu$, and
charge $q$ can be read off from the asymptotic behavior of the fields in
$r\to\infty$ as

$\displaystyle f(r)$ $\displaystyle=r^{2}+1+\cdots-\frac{m}{r^{d-2}}+\cdots\
,$ (88) $\displaystyle A_{t}(r)$
$\displaystyle=\mu+\cdots-\frac{q}{r^{d-2}}+\cdots\ .$ (89)

These are chosen to agree with the notation of $m$, $\mu$, and $q$ used in the
Reissner-Nordstrom black hole solution (4). For $d=4,\,\Delta=4$, we find

$\displaystyle m$
$\displaystyle=\frac{8\kappa^{2}}{9}\epsilon^{2}+\frac{(78336\kappa^{2}-6767e^{2})\kappa^{2}}{39690}\epsilon^{4}+\cdots\
,$ (90) $\displaystyle\mu$
$\displaystyle=\frac{4}{e}+\left(\frac{3e}{14}-\frac{32\kappa^{2}}{21e}\right)\epsilon^{2}+\frac{122400480\kappa^{2}e^{2}-574944256\kappa^{4}-6383817e^{4}}{97796160e}\epsilon^{4}+\cdots\
,$ (91) $\displaystyle q$
$\displaystyle=\frac{e}{6}\epsilon^{2}+\frac{(2658\kappa^{2}-241e^{2})e}{6615}\epsilon^{4}+\cdots\
,$ (92)

where we explicitly included the $O(\epsilon^{4})$ contributions that are
necessary for obtaining $m$ to $O(q^{2})$. In the above expression, we have
used the degrees of freedom to rescale $\epsilon$ to set the coefficient of
$r^{-\Delta}$ in $\phi(r)$ in $r\to\infty$ to be exactly $\epsilon$ (no higher
orders in $\epsilon$): $\phi(r)=\epsilon/r^{4}+\cdots$. From this we can also
read off the scalar condensate (vacuum expectation value), but we will not use
it, so we omit it. Because we wish to express $m$ and $\mu$ perturbatively as
functions of small $q$, we invert $q$ as

$\displaystyle\epsilon=\sqrt{\frac{6}{e}}\,q^{1/2}+\frac{2\sqrt{6}(241e^{2}-2658\kappa^{2})}{735e^{3/2}}q^{3/2}+\cdots\
.$ (93)

Substituting this into $m$ and $\mu$, we obtain

$\displaystyle m$
$\displaystyle=\frac{16\kappa^{2}}{3e}q+\frac{2\kappa^{2}}{21}\left(9-\frac{64\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (94) $\displaystyle\mu$
$\displaystyle=\frac{4}{e}+\left(\frac{9}{7}-\frac{64\kappa^{2}}{7e^{2}}\right)q+\cdots\
.$ (95)

Thus, we reproduce the results for the massless scalar in AdS5 [12].

We also provide some details for massive scalar in asymptotically AdS4 $(d=3)$
with $\Delta=2$ $(m_{\phi}^{2}=-2)$ [16]. We find

$\displaystyle f(r)$
$\displaystyle=1+r^{2}-\frac{r+(1+r^{2})\tan^{-1}r}{r(1+r^{2})}\kappa^{2}\epsilon^{2}+\cdots\
,$ (96) $\displaystyle g(r)$
$\displaystyle=\frac{1}{1+r^{2}}-\frac{r-(1+r^{2})\tan^{-1}r}{r(1+r^{2})^{3}}\kappa^{2}\epsilon^{2}+\cdots\
,$ (97) $\displaystyle A_{t}(r)$
$\displaystyle=\frac{2}{e}+\left(\frac{e(1+5r^{2})}{8(1+r^{2})}-\frac{\kappa^{2}}{2e}-\frac{e\tan^{-1}r}{2r}\right)\epsilon^{2}+\cdots\
,$ (98) $\displaystyle\phi(r)$ $\displaystyle=\frac{\epsilon}{1+r^{2}}+\cdots\
.$ (99)

From these, up to $O(\epsilon^{4})$, we obtain121212Note that our expansion
parameter $\epsilon$ is different from [16], and therefore the coefficients in
(102) are different in higher orders in $\epsilon$. However, the ambiguities
in the parametrization of $\epsilon$ cancel out when $m$ and $\mu$ are written
as functions of $q$ as in (104).

$\displaystyle m$
$\displaystyle=\frac{\pi\kappa^{2}}{2}\epsilon^{2}+\frac{\pi\kappa^{2}}{192}\left((65+4\pi^{2})e^{2}-4(61+4\pi^{2})\kappa^{2}\right)\epsilon^{4}+\cdots\
,$ (100) $\displaystyle\mu$
$\displaystyle=\frac{2}{e}+\left(\frac{5e}{8}-\frac{\kappa^{2}}{2e}\right)\epsilon^{2}+\frac{41e^{4}(2\pi^{2}-15)+4\kappa^{2}e^{2}(661-93\pi^{2})+4\kappa^{4}(98\pi^{2}-821)}{384e}\epsilon^{4}+\cdots\
,$ (101) $\displaystyle q$ $\displaystyle=\frac{\pi
e}{4}\epsilon^{2}+\frac{\pi
e}{192}\left(e^{2}(25+2\pi^{2})-4\kappa^{2}(29+2\pi^{2})\right)\epsilon^{4}+\cdots\
.$ (102)

Rewriting $m$ and $\mu$ as functions of $q$, we find

$\displaystyle m$
$\displaystyle=\frac{2\kappa^{2}}{e}q+\frac{\kappa^{2}}{4\pi}\left(5-\frac{4\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (103) $\displaystyle\mu$
$\displaystyle=\frac{2}{e}+\frac{1}{2\pi}\left(5-\frac{4\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (104)

We repeat the above calculations for various integer values of $\Delta$ in
general dimensions $d$. Some results of $m$ and $\mu$ in small $q$ expansion
are listed below.

  * •

$d=3,\,\Delta=1$ (alternative quantization):

$\displaystyle m$
$\displaystyle=\frac{\kappa^{2}}{e}q+\frac{\kappa^{2}}{2\pi}\left(1+\frac{\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (105) $\displaystyle\mu$
$\displaystyle=\frac{1}{e}+\frac{1}{\pi}\left(1+\frac{\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (106)

  * •

$d=3,\,\Delta=2$:

$\displaystyle m$
$\displaystyle=\frac{2\kappa^{2}}{e}q+\frac{\kappa^{2}}{4\pi}\left(5-\frac{4\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (107) $\displaystyle\mu$
$\displaystyle=\frac{2}{e}+\frac{1}{2\pi}\left(5-\frac{4\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (108)

  * •

$d=3,\,\Delta=3$:

$\displaystyle m$
$\displaystyle=\frac{3\kappa^{2}}{e}q+\frac{7\kappa^{2}}{4\pi}\left(1-\frac{3\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (109) $\displaystyle\mu$
$\displaystyle=\frac{3}{e}+\frac{7}{2\pi}\left(1-\frac{3\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (110)

  * •

$d=4,\,\Delta=2$ (BF bound):

$\displaystyle m$
$\displaystyle=\frac{8\kappa^{2}}{3e}q+\frac{2\kappa^{2}}{9}q^{2}+\cdots\ ,$
(111) $\displaystyle\mu$ $\displaystyle=\frac{2}{e}+\frac{1}{3}q+\cdots\ .$
(112)

  * •

$d=4,\,\Delta=3$:

$\displaystyle m$
$\displaystyle=\frac{4\kappa^{2}}{e}q+\frac{8\kappa^{2}}{15}\left(1-\frac{3\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (113) $\displaystyle\mu$
$\displaystyle=\frac{3}{e}+\frac{4}{5}\left(1-\frac{3\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (114)

  * •

$d=4,\,\Delta=4$:

$\displaystyle m$
$\displaystyle=\frac{16\kappa^{2}}{3e}q+\frac{2\kappa^{2}}{21}\left(9-\frac{64\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (115) $\displaystyle\mu$
$\displaystyle=\frac{4}{e}+\left(\frac{9}{7}-\frac{64\kappa^{2}}{7e^{2}}\right)q+\cdots\
.$ (116)

  * •

$d=5,\,\Delta=2$ (alternative quantization):

$\displaystyle m$
$\displaystyle=\frac{3\kappa^{2}}{e}q+\frac{\kappa^{2}}{8\pi}\left(1+\frac{2\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (117) $\displaystyle\mu$
$\displaystyle=\frac{2}{e}+\frac{1}{6\pi}\left(1+\frac{2\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (118)

  * •

$d=5,\,\Delta=3$:

$\displaystyle m$
$\displaystyle=\frac{9\kappa^{2}}{2e}q+\frac{\kappa^{2}}{16\pi}\left(14-\frac{27\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (119) $\displaystyle\mu$
$\displaystyle=\frac{3}{e}+\frac{1}{12\pi}\left(14-\frac{27\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (120)

  * •

$d=5,\,\Delta=4$:

$\displaystyle m$
$\displaystyle=\frac{6\kappa^{2}}{e}q+\frac{3\kappa^{2}}{16\pi}\left(11-\frac{72\kappa^{2}}{e^{2}}\right)q^{2}+\cdots\
,$ (121) $\displaystyle\mu$
$\displaystyle=\frac{4}{e}+\frac{1}{4\pi}\left(11-\frac{72\kappa^{2}}{e^{2}}\right)q+\cdots\
.$ (122)

By comparing such results for integer values of $\Delta$ in general dimensions
$d$, we find that the general expression for $m$ and $\mu$ of the soliton up
to $O(q^{2})$ is given by

$\displaystyle m$
$\displaystyle=\frac{2(d-2)\Delta\kappa^{2}}{(d-1)e}q+\frac{(d-2)\kappa^{2}\,\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}{(d-1)\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}\left(1-\frac{(d-2)\Delta^{2}(\Delta-d/2)}{(d-1)(\Delta-d/4)}\frac{\kappa^{2}}{e^{2}}\right)q^{2}\
,$ (123) $\displaystyle\mu$
$\displaystyle=\frac{\Delta}{e}+\frac{\Gamma(\Delta)^{2}\,\Gamma(2\Delta+1-d/2)}{\Gamma(d/2)\,\Gamma(2\Delta)\,\Gamma(\Delta+1-d/2)^{2}}\left(1-\frac{(d-2)\Delta^{2}(\Delta-d/2)}{(d-1)(\Delta-d/4)}\frac{\kappa^{2}}{e^{2}}\right)q\
.$ (124)

We note that coefficients diverge when $\Delta=d/4$, which can be realized
only in $d=3$ at $\Delta=3/4$ (in $d\geq 4$, we always have $\Delta>d/2-1\geq
d/4$). This expansion formula, deduced from the results of integer $\Delta$,
hence might not be suitable for $\Delta\leq 3/4$ in $d=3$. At the end of
section 2, we have discussed the same issue from the viewpoint of charge
convexity. To avoid subtleties, in this paper we assume $\Delta>3/4$ when
$d=3$.

##  Appendix D Construction of small charge hairy black holes

The critical coupling $e=e_{c}$ is where the extremal Reissner-Nordstrom black
hole and soliton have the same energy for the same charge. Here, near
$e=e_{c}$, we evaluate the energy of the hairy black hole as a non-interacting
mix of the extremal Reissner-Nordstrom black hole and soliton. This method was
successfully employed for massless scalar in $d=4$ in [14], and here we
generalize the calculations to other $\Delta$ and $d$.

###  D.1 Construction in $e<e_{c}$

In $e<e_{c}$, the Reissner-Nordstrom black hole has a lower energy than the
soliton around $Q=Q_{c}$. At $Q=Q_{c}$, we start to mix small charge soliton
contributions to the Reissner-Nordstrom black hole to evaluate the energy of a
hairy black hole. We will see that the Reissner-Nordstrom black hole is the
lowest energy state in $Q<Q_{c}$, while the hairy black hole is the lowest
energy state in $Q>Q_{c}$.

In $e<e_{c}$, we express the small difference of the coupling $e$ from $e_{c}$
by introducing a small parameter $\theta$ as

$\displaystyle e^{2}=e_{c}^{2}(1-\theta)\ .$ (125)

From (64), the chemical potential and charge of the extremal Reissner-
Nordstrom black hole depends on small $r_{h}$,

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{1}{\kappa}\sqrt{\frac{d-1}{d-2}}\left(1+\frac{d}{2(d-2)}r_{h}^{2}+O(r_{h}^{4})\right)\
,$ (126) $\displaystyle q_{RN}$
$\displaystyle=\frac{r_{h}^{d-2}}{\kappa}\sqrt{\frac{d-1}{d-2}}\left(1+\frac{d}{2(d-2)}r_{h}^{2}+O(r_{h}^{4})\right)\
.$ (127)

To mix the Reissner-Nordstrom black hole and soliton, we parametrize $r_{h}$
and the charge of the soliton as perturbative series of $\theta$ so that they
can be compared at each order in the perturbative series. In $e<e_{c}$, we
introduce the ansatz that small soliton contributions are put on top of the
Reissner-Nordstrom black hole. Hence, we devise the perturbative series so
that $\mu_{RN}$ and $q_{RN}$ are larger than $\mu_{sol}$ and $q_{sol}$.

####  D.1.1 $d=3$

To mix the Reissner-Nordstrom black hole and soliton for $d=3$ in $e<e_{c}$,
we take $q_{RN}\sim r_{h}\sim\theta^{1/2}$. We use the following ansatz for
$r_{h}$,

$\displaystyle
r_{h}=a_{1}\theta^{1/2}+a_{2}\theta+a_{3}\theta^{3/2}+a_{4}\theta^{2}+\cdots\
,$ (128)

where we have explicitly included the terms necessary for calculating the
hairy black hole energy up to $O(q^{2})$. With this ansatz, $\mu_{RN}$,
$q_{RN}$, and $m_{RN}$ of the Reissner-Nordstrom black hole can be expanded in
powers of $\theta$ as

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{\sqrt{2}}{\kappa}+\frac{3a_{1}^{2}}{\sqrt{2}\kappa}\theta+\frac{3\sqrt{2}a_{1}a_{2}}{\kappa}\theta^{3/2}+\frac{3(3a_{1}^{4}-4a_{2}^{2}-8a_{1}a_{3})}{4\sqrt{2}\kappa}\theta^{2}+\frac{-9a_{1}^{3}a_{2}+6a_{2}a_{3}+6a_{1}a_{4}}{\sqrt{2}\kappa}\theta^{5/2}+\cdots\
,$ (129) $\displaystyle q_{RN}$
$\displaystyle=\frac{\sqrt{2}a_{1}}{\kappa}\theta^{1/2}+\frac{\sqrt{2}a_{2}}{\kappa}\theta+\frac{\sqrt{2}(3a_{1}^{3}+2a_{3})}{2\kappa}\theta^{3/2}+\frac{\sqrt{2}(9a_{1}^{2}a_{2}+2a_{4})}{2\kappa}\theta^{2}+\cdots\
,$ (130) $\displaystyle m_{RN}$
$\displaystyle=2a_{1}\theta^{1/2}+2a_{2}\theta+2(2a_{1}^{3}+a_{3})\theta^{3/2}+(6a_{1}^{2}a_{2}+a_{4})\theta^{2}+\cdots\
.$ (131)

To balance the chemical potential, we take the ansatz for the charge of the
soliton as

$\displaystyle q_{sol}=b_{1}\theta+b_{2}\theta^{3/2}+b_{3}\theta^{2}+\cdots\
.$ (132)

Here, the leading behavior of the soliton is chosen as $q_{sol}\sim\theta$ in
order to balance the chemical potential. Then, the chemical potential and mass
of the soliton can be expressed as

$\displaystyle\mu_{sol}$
$\displaystyle=\frac{\sqrt{2}}{\kappa}+\left(\frac{1}{\sqrt{2}\kappa}+\frac{3b_{1}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\right)\theta+\frac{3b_{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\theta^{3/2}$
(133)
$\displaystyle\qquad+\left(\frac{3}{4\sqrt{2}\kappa}+\frac{(3b_{3}+2b_{1}(3-2\Delta))\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\right)\theta^{2}+\cdots\
,$ (134) $\displaystyle m_{sol}$
$\displaystyle=\sqrt{2}b_{1}\kappa\theta+\sqrt{2}b_{2}\kappa\theta^{3/2}+\left(\frac{(b_{1}+2b_{3})\kappa}{\sqrt{2}}+\frac{3b_{1}^{2}\kappa^{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\right)\theta^{2}+\cdots\
,$ (135)

where $b_{1}>0$ is assumed so that $m_{sol}>0$. Note that, with the above
ansatz, the charge of the hairy black hole has the series expansion of the
form

$\displaystyle
q=q_{RN}+q_{sol}=\frac{\sqrt{2}a_{1}}{\kappa}\theta^{1/2}+\left(\frac{\sqrt{2}a_{2}}{\kappa}+b_{1}\right)\theta+\cdots\
.$ (136)

At this point, we fix the redefinition ambiguities in the $\theta$ expansion
by demanding $q\sim\theta^{1/2}$ with no higher $\theta$ corrections.131313In
[14], a different choice to fix the ambiguities was made (i.e. $q_{sol}$ has
no higher order $\theta$ corrections), but eventually, if we express
$m_{hairy}$ as a function of $q$, the ambiguities will be gone. Then, each
term in the above expansion should satisfy

$\displaystyle q=\frac{\sqrt{2}a_{1}}{\kappa}\theta^{1/2}\
,\qquad\frac{2a_{2}}{\kappa}+b_{1}=0\ ,$ (137)

and so on for the higher orders involving $a_{3,4}$ and $b_{2,3}$. The
balancing of the chemical potential $\mu_{RN}=\mu_{sol}$ also gives the
equations

$\displaystyle\frac{3a_{1}^{2}}{\sqrt{2}\kappa}$
$\displaystyle=\frac{1}{\sqrt{2}\kappa}+\frac{3b_{1}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\
,$ (138) $\displaystyle\frac{3\sqrt{2}a_{1}a_{2}}{\kappa}$
$\displaystyle=\frac{3b_{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\
,$ (139)

and so on for $a_{3}$ and $b_{3}$. From these equations, we can determine the
coefficients $a_{1,2,3,4}$ and $b_{1,2,3}$. The upshot is that, in $q>q_{c}$,
the mass of the hairy black hole $m_{hairy}=m_{RN}+m_{sol}$ is given by

$\displaystyle m_{hairy}=m_{RN}-\frac{3\times
2^{2\Delta-5}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\kappa^{4}(q^{2}-q_{c}^{2})^{2}\
,$ (140)

where

$\displaystyle q_{c}$ $\displaystyle=\frac{\sqrt{6}}{3\kappa}\theta^{1/2}\ ,$
(141) $\displaystyle m_{RN}$ $\displaystyle=\sqrt{2}\kappa
q+\frac{\kappa^{3}q^{3}}{2\sqrt{2}}+O(q^{5})\ .$ (142)

Recall we assume $\Delta>3/4$. Then, from (140), we see that
$m_{hairy}<m_{RN}$. Note that we also find $b_{1}\to 0$ in $q\to q_{c}$ from
above.

In fact, we find $q_{sol}<0$ in $q<q_{c}$, and therefore the above expression
cannot be applied in $q<q_{c}$, where $m_{sol}$ would be $m_{sol}<0$ if
$m_{sol}=q_{sol}+\cdots$ is assumed. This means that we need to parametrize
the soliton mass as $m_{sol}=|q_{sol}|+\cdots$. Then, repeating the
calculations, we find that the mass of the hairy black hole in $q<q_{c}$ is
given by

$\displaystyle
m_{hairy}=m_{RN}+\frac{2^{2\Delta-1}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\kappa^{2}(q_{c}^{2}-q^{2})\
.$ (143)

Thus, $m_{hairy}>m_{RN}$ in $q<q_{c}$ if $\Delta>3/4$. This is reasonable
because in $q<q_{c}$ this construction attempts to approximate the hairy black
hole as a non-interacting mix of the Reissner-Nordstrom black hole and soliton
with the opposite charges.

####  D.1.2 $d=4$

In $d=4$, we use $q_{RN}\sim r_{h}^{2}\sim\theta$. The ansatz for $r_{h}$ is

$\displaystyle r_{h}^{2}=a_{1}\theta+a_{2}\theta^{2}+\cdots\ .$ (144)

With this ansatz, $\mu_{RN}$, $q_{RN}$, and $m_{RN}$ takes the form

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{\sqrt{6}}{2\kappa}+\frac{\sqrt{6}a_{1}}{2\kappa}\theta+\frac{\sqrt{6}(2a_{2}-a_{1}^{2})}{4\kappa}\theta^{2}+\cdots\
,$ (145) $\displaystyle q_{RN}$
$\displaystyle=\frac{\sqrt{6}a_{1}}{2\kappa}\theta+\frac{\sqrt{6}(a_{1}^{2}+a_{2})}{2\kappa}\theta^{2}+\cdots\
,$ (146) $\displaystyle m_{RN}$
$\displaystyle=2a_{1}\theta+(3a_{1}^{2}+2a_{2})\theta^{2}+\cdots\ .$ (147)

To match this, we use the ansatz for the charge of the soliton as

$\displaystyle q_{sol}=b_{1}\theta+b_{2}\theta^{2}+\cdots\ .$ (148)

The chemical potential and mass of the soliton are

$\displaystyle\mu_{sol}$
$\displaystyle=\frac{\sqrt{6}}{2\kappa}+\left(\frac{b_{1}(\Delta-1)}{2\Delta-1}+\frac{\sqrt{6}}{4\kappa}\right)\theta+\left(\frac{(\Delta-1)(b_{2}-b_{1}(\Delta-2))}{2\Delta-1}+\frac{3\sqrt{6}}{16\kappa}\right)\theta^{2}+\cdots\
,$ (149) $\displaystyle m_{sol}$ $\displaystyle=\frac{2\sqrt{6}\kappa
b_{1}}{3}\theta+\frac{1}{3}\left(\sqrt{6}\kappa(b_{1}+2b_{2})+\frac{2\kappa^{2}b_{1}^{2}(\Delta-1)}{2\Delta-1}\right)\theta^{2}+\cdots\
,$ (150)

where $b_{1}>0$. The charge of the hairy black hole has the series expansion
of the form

$\displaystyle
q=q_{RN}+q_{sol}=\left(\frac{\sqrt{6}a_{1}}{2\kappa}+b_{1}\right)\theta+\left(\frac{\sqrt{6}(a_{1}^{2}+a_{2})}{2\kappa}+b_{2}\right)\theta^{2}+\cdots\
.$ (151)

Solving $\mu_{RN}=\mu_{sol}$ and fixing the redefinition ambiguities by
demanding $q\sim\theta$ with no higher $\theta$ corrections, we can determine
$a_{1,2}$ and $b_{1,2}$. The mass $m_{hairy}=m_{RN}+m_{sol}$ is obtained in
$q>q_{c}$ as

$\displaystyle
m_{hairy}=m_{RN}-\frac{2(2\Delta-1)}{3(3\Delta-2)}\kappa^{2}(q-q_{c})^{2}\ ,$
(152)

where

$\displaystyle q_{c}$ $\displaystyle=\frac{\sqrt{6}}{4\kappa}\theta\ ,$ (153)
$\displaystyle m_{RN}$ $\displaystyle=\frac{2\sqrt{6}}{3}\kappa
q+\frac{2}{3}\kappa^{2}q^{2}\ .$ (154)

In $q<q_{c}$, where $q_{sol}<0$, we find

$\displaystyle
m_{hairy}=m_{RN}+\frac{4\sqrt{6}(2\Delta-1)\kappa}{3(3\Delta-2)}(q_{c}-q)>m_{RN}\
.$ (155)

####  D.1.3 $d=5$

In $d=5$, we use $q_{RN}\sim r_{h}^{3}\sim\theta^{3/2}$. The ansatz for
$r_{h}$ we choose is

$\displaystyle
r_{h}=a_{1}\theta^{1/2}+a_{2}\theta+a_{3}\theta^{3/2}+a_{4}\theta^{2}+\cdots\
.$ (156)

With this ansatz, $\mu_{RN}$, $q_{RN}$, and $m_{RN}$ can be parametrized as

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{2}{\sqrt{3}\kappa}+\frac{5a_{1}^{2}}{3\sqrt{3}\kappa}\theta+\frac{10a_{1}a_{2}}{3\sqrt{3}\kappa}\theta^{3/2}+\frac{5(-5a_{1}^{4}+12a_{2}^{2}+24a_{1}a_{3})}{36\sqrt{3}\kappa}\theta^{2}$
(157)
$\displaystyle\qquad+\frac{5(-5a_{1}^{3}a_{2}+6a_{2}a_{3}+6a_{1}a_{4})}{9\sqrt{3}\kappa}\theta^{5/2}+\cdots\
,$ (158) $\displaystyle q_{RN}$
$\displaystyle=\frac{2a_{1}^{3}}{\sqrt{3}\kappa}\theta^{3/2}+\frac{2\sqrt{3}a_{1}^{2}a_{2}}{\kappa}\theta^{2}+\frac{5a_{1}^{5}+18a_{1}a_{2}^{2}+18a_{1}^{2}a_{3}}{3\sqrt{3}\kappa}\theta^{5/2}$
(159)
$\displaystyle\qquad+\frac{25a_{1}^{4}a_{2}+6a_{2}^{3}+36a_{1}a_{2}a_{3}+18a_{1}^{2}a_{4}}{3\sqrt{3}\kappa}\theta^{3}+\cdots\
,$ (160) $\displaystyle m_{RN}$
$\displaystyle=2a_{1}^{3}\theta^{3/2}+6a_{1}^{2}a_{2}\theta^{2}+\left(\frac{8a_{1}^{5}}{3}+6a_{1}a_{2}^{2}+6a_{1}^{2}a_{3}\right)\theta^{5/2}$
(161)
$\displaystyle\qquad+\left(\frac{40a_{1}^{4}a_{2}}{3}+2a_{2}^{3}+12a_{1}a_{2}a_{3}+6a_{1}^{4}a_{4}\right)\theta^{3}+\cdots\
.$ (162)

For the matching of the scaling of the above charge, we take the ansatz for
the charge of the soliton as

$\displaystyle
q_{sol}=b_{1}\theta^{3/2}+b_{2}\theta^{2}+b_{3}\theta^{5/2}+\cdots\ .$ (163)

The chemical potential and mass of the soliton are

$\displaystyle\mu_{sol}$
$\displaystyle=\frac{2}{\sqrt{3}\kappa}+\frac{1}{\sqrt{3}\kappa}\theta+\frac{5\times
2^{2-2\Delta}b_{1}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{3\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\theta^{3/2}$
(164) $\displaystyle\qquad+\left(\frac{\sqrt{3}}{4\kappa}+\frac{5\times
2^{2-2\Delta}b_{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{3\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\right)\theta^{2}$
(165)
$\displaystyle\qquad+\frac{2^{2-2\Delta}(5b_{3}-2b_{1}(2\Delta-5))\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{3\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\theta^{5/2}+\cdots\
,$ (166) $\displaystyle m_{sol}$ $\displaystyle=\sqrt{3}\kappa
b_{1}\theta^{3/2}+\sqrt{3}\kappa
b_{2}\theta^{2}+\frac{\sqrt{3}}{2}\kappa(2b_{3}+b_{1})\theta^{5/2}+\cdots\ ,$
(167)

where $b_{1}>0$. The charge of the hairy black hole has the series expansion
of the form

$\displaystyle
q=q_{RN}+q_{sol}=\left(\frac{2a_{1}^{3}}{\sqrt{3}\kappa}+b_{1}\right)\theta^{3/2}+\left(\frac{2\sqrt{3}a_{1}^{2}a_{2}}{\kappa}+b_{2}\right)\theta^{2}+\cdots\
.$ (168)

We solve $\mu_{RN}=\mu_{sol}$ and $q\sim\theta^{3/2}$ for $a_{1,2,3,4}$ and
$b_{1,2,3}$. The mass $m_{hairy}=m_{RN}+m_{sol}$ is then given in $q>q_{c}$ by

$\displaystyle m_{hairy}$ $\displaystyle=\sqrt{3}\kappa
q+\left(\frac{\sqrt{3}}{2}\kappa
q\theta-\frac{6\sqrt{15}}{125}\theta^{5/2}\right)$ (169)
$\displaystyle=m_{RN}-\left(\frac{3^{5/6}\kappa^{5/3}q^{5/3}}{2^{5/3}}-\frac{\sqrt{3}}{2}\kappa
q\theta+\frac{6\sqrt{15}}{125}\theta^{5/2}\right)\ ,$ (170)

where

$\displaystyle q_{c}$ $\displaystyle=\frac{6}{5\sqrt{5}\kappa}\theta^{3/2}\ ,$
(171) $\displaystyle m_{RN}$ $\displaystyle=\sqrt{3}\kappa
q+\frac{3^{5/6}\kappa^{5/3}q^{5/3}}{2^{5/3}}\ .$ (172)

In $q<q_{c}$, where $q_{sol}<0$, we find

$\displaystyle m_{hairy}=m_{RN}+2\sqrt{3}\kappa(q_{c}-q)>m_{RN}\ .$ (173)

####  D.1.4 $d\geq 6$

In $e<e_{c}$, hairy black holes for $d\geq 6$ can be constructed in a similar
way as $d=5$ and are of no qualitative difference. We first notice $q_{RN}\sim
r_{h}^{d-2}\sim\theta^{(d-2)/2}$ (see (28) and (64)). Then, we parametrize
$r_{h}$ and $q_{sol}$ by series expansion in $\theta$ so that $q_{RN}$ and
$q_{sol}$ have the same powers of $\theta$. To calculate the energy up to
$O(q^{2})$, for even $d$, we use the ansatz

$\displaystyle r_{h}^{2}$
$\displaystyle=a_{1}\theta+a_{2}\theta^{2}+\cdots+a_{d/2}\theta^{{d/2}}+\cdots\
,$ (174) $\displaystyle q_{sol}$
$\displaystyle=b_{1}\theta^{(d-2)/2}+b_{2}\theta^{(d-2)/2+1}+\cdots+b_{d/2}\theta^{d-2}+\cdots\
.$ (175)

and for odd $d$, we alternatively use the ansatz

$\displaystyle r_{h}$
$\displaystyle=a_{1}\theta^{1/2}+a_{2}\theta+\cdots+a_{d-1}\theta^{{(d-1)/2}}+\cdots\
,$ (176) $\displaystyle q_{sol}$
$\displaystyle=b_{1}\theta^{(d-2)/2}+b_{2}\theta^{(d-2)/2+1/2}+\cdots+b_{d-2}\theta^{d-2}+\cdots\
.$ (177)

For example, for $d=6$, we have

$\displaystyle r_{h}^{2}$
$\displaystyle=a_{1}\theta+a_{2}\theta^{2}+a_{3}\theta^{3}+\cdots\ ,$ (178)
$\displaystyle q_{sol}$
$\displaystyle=b_{1}\theta^{2}+b_{2}\theta^{3}+b_{3}\theta^{4}+\cdots\ .$
(179)

As we have done above, we can determine parameters $a_{i}$ and $b_{i}$ by
balancing the chemical potential $\mu_{RN}=\mu_{sol}$ and demanding the charge
to be $q\sim\theta^{(d-2)/2}$ (with no higher order corrections). We can
systematically calculate the energy of the hairy black hole for any $d\geq 6$.
Here, we list some results.

  * •

$d=6$: We have $q\sim\theta^{2}$. The phase transition at $q=q_{c}$ is
continuous as

$\displaystyle m_{hairy}$
$\displaystyle=m_{RN}-\left(\frac{2^{3/2}\kappa^{3/2}q^{3/2}}{5^{3/4}}-\frac{2}{\sqrt{5}}\kappa
q\theta+\frac{4}{27}\theta^{3}\right)$ (180)
$\displaystyle=m_{RN}-\frac{9\kappa^{2}}{20\,\theta}(q-q_{c})^{2}+O\left((q-q_{c})^{3}\right)\
,$ (181)

where the second line is the Taylor expansion around $q=q_{c}$, where
$|q-q_{c}|\ll\theta^{2}$, and

$\displaystyle m_{RN}$ $\displaystyle=\frac{4}{\sqrt{5}}\kappa
q+\frac{2^{3/2}\kappa^{3/2}q^{3/2}}{5^{3/4}}+\cdots\ ,$ (182) $\displaystyle
q_{c}$ $\displaystyle=\frac{2\sqrt{5}}{9\kappa}\theta^{2}\ .$ (183)

By considering the $O(q^{2})$ terms as well, we can check the convexity
(because $\Delta>2$) as

$\displaystyle\frac{d^{2}m_{hairy}}{dq^{2}}=\frac{6(\Delta-1)(\Delta-2)^{2}\kappa^{2}}{5(2\Delta-1)(2\Delta-3)}>0\
.$ (184)

  * •

$d=7$: We have $q\sim\theta^{5/2}$. The phase transition at $q=q_{c}$ is
continuous as

$\displaystyle m_{hairy}$
$\displaystyle=m_{RN}-\left(\frac{5^{7/10}\kappa^{7/5}q^{7/5}}{6^{7/10}}-\sqrt{\frac{5}{6}}\,\kappa
q\theta+\frac{2\times 5^{5/2}}{7^{7/2}}\theta^{7/2}\right)$ (185)
$\displaystyle=m_{RN}-\frac{7^{5/2}\kappa^{2}}{6\times
5^{5/2}\theta^{3/2}}(q-q_{c})^{2}+O\left((q-q_{c})^{3}\right)\ ,$ (186)

where the second line is the Taylor expansion around $q=q_{c}$, where
$|q-q_{c}|\ll\theta^{5/2}$, and

$\displaystyle m_{RN}$ $\displaystyle=\sqrt{\frac{10}{3}}\,\kappa
q+\frac{5^{7/10}\kappa^{7/5}q^{7/5}}{6^{7/10}}+\cdots\ ,$ (187) $\displaystyle
q_{c}$ $\displaystyle=\frac{25\sqrt{6}}{7^{5/2}\kappa}\theta^{5/2}\ .$ (188)

By considering the $O(q^{2})$ terms as well, we can check the convexity
(because $\Delta>5/2$) as

$\displaystyle\frac{d^{2}m_{hairy}}{dq^{2}}=\frac{7\kappa^{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-7/2)}{9\times
2^{2\Delta-3}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-5/2)^{2}}>0\ .$ (189)

  * •

$d=8$: We have $q\sim\theta^{3}$. The phase transition at $q=q_{c}$ is
continuous as

$\displaystyle m_{hairy}$
$\displaystyle=m_{RN}-\left(\frac{6^{2/3}\kappa^{4/3}q^{4/3}}{7^{2/3}}-\sqrt{\frac{6}{7}}\,\kappa
q\theta+\frac{27}{256}\theta^{4}\right)$ (190)
$\displaystyle=m_{RN}-\frac{64\kappa^{2}}{189\,\theta^{2}}(q-q_{c})^{2}+O\left((q-q_{c})^{3}\right)\
,$ (191)

where the second line is the Taylor expansion around $q=q_{c}$, where
$|q-q_{c}|\ll\theta^{3}$, and

$\displaystyle m_{RN}$ $\displaystyle=2\sqrt{\frac{6}{7}}\,\kappa
q+\frac{6^{2/3}\kappa^{4/3}q^{4/3}}{7^{2/3}}+\cdots\ ,$ (192) $\displaystyle
q_{c}$ $\displaystyle=\frac{9\sqrt{42}}{128\kappa}\theta^{3}\ .$ (193)

By considering the $O(q^{2})$ terms as well, we can check the convexity
(because $\Delta>3$) as

$\displaystyle\frac{d^{2}m_{hairy}}{dq^{2}}=\frac{2(\Delta-1)(\Delta-2)(\Delta-3)^{2}\kappa^{2}}{7(2\Delta-1)(2\Delta-3)}>0\
.$ (194)

###  D.2 Construction in $e>e_{c}$

In $e>e_{c}$, the soliton has a lower energy than the Reissner-Nordstrom black
hole around $Q=Q_{c}$. At $Q=Q_{c}$, we start to mix small charge Reissner-
Nordstrom black hole contributions to the soliton to obtain a hairy black
hole. We will see that the soliton is the lowest energy state in $Q<Q_{c}$,
while the hairy black hole is the lowest energy state in $Q>Q_{c}$.

In $e>e_{c}$, we write the small difference of the coupling $e$ from $e_{c}$
as

$\displaystyle e^{2}=e_{c}^{2}(1+\theta)\ .$ (195)

####  D.2.1 $d=3$

In $d=3$ and $e>e_{c}$, we take $q_{RN}\sim r_{h}\sim\theta$ so that $q_{RN}$
is not more dominant than $q_{sol}$. The ansatz for $r_{h}$ can be taken as

$\displaystyle r_{h}=a_{1}\theta+a_{2}\theta^{2}+\cdots\ .$ (196)

With this ansatz, $\mu_{RN}$, $q_{RN}$, and $m_{RN}$ can be parametrized as

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{\sqrt{2}}{\kappa}+\frac{3a_{1}^{2}}{\sqrt{2}\kappa}\theta^{2}+\cdots\
,$ (197) $\displaystyle q_{RN}$
$\displaystyle=\frac{\sqrt{2}a_{1}}{\kappa}\theta+\frac{\sqrt{2}a_{2}}{\kappa}\theta^{2}+\cdots\
,$ (198) $\displaystyle m_{RN}$
$\displaystyle=2a_{1}\theta+2a_{2}\theta^{2}+\cdots\ .$ (199)

To match this, we use the ansatz for the charge of the soliton as

$\displaystyle q_{sol}=b_{1}\theta+b_{2}\theta^{2}+\cdots\ .$ (200)

The chemical potential and mass of the soliton are then

$\displaystyle\mu_{sol}$
$\displaystyle=\frac{\sqrt{2}}{\kappa}+\left(-\frac{1}{\sqrt{2}\kappa}+\frac{3b_{1}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\right)\theta+$
(201)
$\displaystyle\qquad+\left(\frac{3}{4\sqrt{2}\kappa}+\frac{(3b_{2}+2b_{1}(2\Delta-3))\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta-1}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\right)\theta^{2}+\cdots\
,$ (202) $\displaystyle m_{sol}$
$\displaystyle=\sqrt{2}b_{1}\kappa\theta+\left(\frac{(2b_{2}-b_{1})\kappa}{\sqrt{2}}+\frac{3b_{1}^{2}\kappa^{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-1/2)^{2}}\right)\theta^{2}+\cdots\
,$ (203)

where $b_{1}>0$. The charge of the hairy black hole takes the form

$\displaystyle
q=q_{RN}+q_{sol}=\left(\frac{\sqrt{2}a_{1}}{\kappa}+b_{1}\right)\theta+\left(\frac{\sqrt{2}a_{2}}{\kappa}+b_{2}\right)\theta^{2}+\cdots\
.$ (204)

We solve $\mu_{RN}=\mu_{sol}$ and $q\sim\theta$ for $a_{1,2}$ and $b_{1,2}$.
The mass $m_{hairy}=m_{RN}+m_{sol}$ is then given in $q>q_{c}$ as

$\displaystyle m_{hairy}$ $\displaystyle=\sqrt{2}\kappa
q-\frac{2^{2\Delta-3}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{3\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\theta^{2}$
(205)
$\displaystyle=m_{sol}-\frac{3\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}\kappa^{2}(q-q_{c})^{2}\
,$ (206)

where

$\displaystyle q_{c}$
$\displaystyle=\frac{2^{2\Delta-3/2}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}{3\kappa\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}\theta\
,$ (207) $\displaystyle m_{sol}$ $\displaystyle=\sqrt{2}\kappa
q+\left(\frac{3\,\Gamma(\Delta)\,\Gamma(2\Delta-3/2)}{2^{2\Delta}\,\Gamma(\Delta-1/2)^{2}\,\Gamma(\Delta+1/2)}\kappa^{2}q^{2}-\frac{\kappa
q\theta}{\sqrt{2}}\right)\ .$ (208)

In $q<q_{c}$, we find $q_{RN}<0$, and hence we repeat the calculations by
using the expression of the mass to be $m_{RN}=2|a_{1}|\theta+\cdots$. Then,
we obtain

$\displaystyle m_{hairy}=m_{sol}+2\sqrt{2}\kappa(q_{c}-q)>m_{sol}\ .$ (209)

####  D.2.2 $d=4$

In $d=4$, we use $q_{RN}\sim r_{h}^{2}\sim\theta$. The ansatz for $r_{h}$ is

$\displaystyle r_{h}^{2}=a_{1}\theta+a_{2}\theta^{2}+\cdots\ .$ (210)

With this ansatz, $\mu_{RN}$, $q_{RN}$, and $m_{RN}$ are given by

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{\sqrt{6}}{2\kappa}+\frac{\sqrt{6}a_{1}}{2\kappa}\theta+\frac{\sqrt{6}(2a_{2}-a_{1}^{2})}{4\kappa}\theta^{2}+\cdots\
,$ (211) $\displaystyle q_{RN}$
$\displaystyle=\frac{\sqrt{6}a_{1}}{2\kappa}\theta+\frac{\sqrt{6}(a_{1}^{2}+a_{2})}{2\kappa}\theta^{2}+\cdots\
,$ (212) $\displaystyle m_{RN}$
$\displaystyle=2a_{1}\theta+(3a_{1}^{2}+2a_{2})\theta^{2}+\cdots\ .$ (213)

To match this, we use the ansatz for the charge of the soliton as

$\displaystyle q_{sol}=b_{1}\theta+b_{2}\theta^{2}+\cdots\ .$ (214)

The chemical potential and mass of the soliton are

$\displaystyle\mu_{sol}$
$\displaystyle=\frac{\sqrt{6}}{2\kappa}+\left(\frac{b_{1}(\Delta-1)}{2\Delta-1}-\frac{\sqrt{6}}{4\kappa}\right)\theta+\left(\frac{(\Delta-1)(b_{1}(\Delta-2)+b_{2})}{2\Delta-1}+\frac{3\sqrt{6}}{16\kappa}\right)\theta^{2}+\cdots\
,$ (215) $\displaystyle m_{sol}$ $\displaystyle=\frac{2\sqrt{6}\kappa
b_{1}}{3}\theta+\frac{1}{3}\left(\sqrt{6}\kappa(2b_{2}-b_{1})+\frac{2\kappa^{2}b_{1}^{2}(\Delta-1)}{2\Delta-1}\right)\theta^{2}+\cdots\
.$ (216)

The charge of the hairy black hole is

$\displaystyle
q=q_{RN}+q_{sol}=\left(\frac{\sqrt{6}a_{1}}{2\kappa}+b_{1}\right)\theta+\left(\frac{\sqrt{6}(a_{1}^{2}+a_{2})}{2\kappa}+b_{2}\right)\theta^{2}+\cdots\
.$ (217)

We solve $\mu_{RN}=\mu_{sol}$ and $q\sim\theta$ for $a_{1,2}$ and $b_{1,2}$.
Then, we obtain the mass $m_{hairy}=m_{RN}+m_{sol}$ as a function of $q$. The
result is, in $q>q_{c}$,

$\displaystyle
m_{hairy}=m_{sol}-\frac{2(\Delta-1)^{2}}{3(2\Delta-1)(3\Delta-2)}\kappa^{2}(q-q_{c})^{2}\
,$ (218)

where

$\displaystyle q_{c}$
$\displaystyle=\frac{\sqrt{6}(2\Delta-1)}{4(\Delta-1)\kappa}\theta\ ,$ (219)
$\displaystyle m_{sol}$ $\displaystyle=\frac{2\sqrt{6}}{3}\kappa
q+\left(\frac{2(\Delta-1)}{3(2\Delta-1)}\kappa^{2}q^{2}-\frac{\sqrt{6}}{3}q\theta\right)\
.$ (220)

In $q<q_{c}$, where $q_{RN}<0$, we find

$\displaystyle
m_{hairy}=m_{sol}+\frac{4\sqrt{6}(\Delta-1)\kappa}{3(3\Delta-2)}(q_{c}-q)>m_{sol}\
.$ (221)

####  D.2.3 $d=5$

In $d=5$, we use $q_{RN}\sim r_{h}^{3}\sim\theta^{3/2}$. The ansatz for
$r_{h}$ we choose is

$\displaystyle
r_{h}=a_{1}\theta^{1/2}+a_{2}\theta+a_{3}\theta^{3/2}+a_{4}\theta^{2}+\cdots\
.$ (222)

With this ansatz, $\mu_{RN}$, $q_{RN}$, and $m_{RN}$ can be parametrized as

$\displaystyle\mu_{RN}$
$\displaystyle=\frac{2}{\sqrt{3}\kappa}+\frac{5a_{1}^{2}}{3\sqrt{3}\kappa}\theta+\frac{10a_{1}a_{2}}{3\sqrt{3}\kappa}\theta^{3/2}+\frac{5(-5a_{1}^{4}+12a_{2}^{2}+24a_{1}a_{3})}{36\sqrt{3}\kappa}\theta^{2}$
(223)
$\displaystyle\qquad+\frac{5(-5a_{1}^{3}a_{2}+6a_{2}a_{3}+6a_{1}a_{4})}{9\sqrt{3}\kappa}\theta^{5/2}+\cdots\
,$ (224) $\displaystyle q_{RN}$
$\displaystyle=\frac{2a_{1}^{3}}{\sqrt{3}\kappa}\theta^{3/2}+\frac{2\sqrt{3}a_{1}^{2}a_{2}}{\kappa}\theta^{2}+\frac{5a_{1}^{5}+18a_{1}a_{2}^{2}+18a_{1}^{2}a_{3}}{3\sqrt{3}\kappa}\theta^{5/2}$
(225)
$\displaystyle\qquad+\frac{25a_{1}^{4}a_{2}+6a_{2}^{3}+36a_{1}a_{2}a_{3}+18a_{1}^{2}a_{4}}{3\sqrt{3}\kappa}\theta^{3}+\cdots\
,$ (226) $\displaystyle m_{RN}$
$\displaystyle=2a_{1}^{3}\theta^{3/2}+6a_{1}^{2}a_{2}\theta^{2}+\left(\frac{8a_{1}^{5}}{3}+6a_{1}a_{2}^{2}+6a_{1}^{2}a_{3}\right)\theta^{5/2}$
(227)
$\displaystyle\qquad+\left(\frac{40a_{1}^{4}a_{2}}{3}+2a_{2}^{3}+12a_{1}a_{2}a_{3}+6a_{1}^{4}a_{4}\right)\theta^{3}+\cdots\
.$ (228)

For the balancing of the chemical potential, the ansatz for the charge of the
soliton is chosen as

$\displaystyle
q_{sol}=b_{1}\theta+b_{2}\theta^{3/2}+b_{3}\theta^{2}+b_{4}\theta^{5/2}+\cdots\
.$ (229)

The chemical potential and mass of the soliton are

$\displaystyle\mu_{sol}$
$\displaystyle=\frac{2}{\sqrt{3}\kappa}+\left(-\frac{1}{\sqrt{3}\kappa}+\frac{5\times
2^{2-2\Delta}b_{1}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{3\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\right)\theta$
(230) $\displaystyle\qquad+\frac{5\times
2^{2-2\Delta}b_{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{3\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\theta^{3/2}$
(231)
$\displaystyle\qquad+\left(\frac{\sqrt{3}}{4\kappa}+\frac{2^{2-2\Delta}(5b_{3}+2b_{1}(2\Delta-5))\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{3\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\right)\theta^{2}+\cdots\
,$ (232) $\displaystyle m_{sol}$ $\displaystyle=\sqrt{3}\kappa
b_{1}\theta+\sqrt{3}\kappa
b_{2}\theta^{3/2}+\left(\frac{\sqrt{3}}{2}\kappa(2b_{3}-b_{1})+\frac{5b_{1}^{2}\kappa^{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{2^{2\Delta}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\right)\theta^{2}$
(233)
$\displaystyle\qquad+\frac{1}{2}\left(\sqrt{3}\kappa(2b_{4}-b_{2})+\frac{5b_{1}b_{2}\kappa^{2}\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{2^{2\Delta-2}\,\Gamma(\Delta+1/2)\,\Gamma(\Delta-3/2)^{2}}\right)\theta^{5/2}+\cdots\
,$ (234)

where $b_{1}>0$. The charge of the hairy black hole is

$\displaystyle
q=q_{RN}+q_{sol}=b_{1}\theta+\left(\frac{2a_{1}^{3}}{\sqrt{3}\kappa}+b_{2}\right)\theta^{3/2}+\cdots\
.$ (235)

Solving $\mu_{RN}=\mu_{sol}$ and $q\sim\theta$, we obtain $a_{1,2,3,4}$ and
$b_{1,2,3,4}$. The mass $m_{hairy}=m_{RN}+m_{sol}$ is then expressed as the
following function of $q$, in $q>q_{c}$,

$\displaystyle
m_{hairy}=m_{sol}-\frac{3^{1/4}\,\Gamma(\Delta)^{5/2}\,\Gamma(2\Delta-5/2)^{5/2}}{2^{5\Delta-6}\,\Gamma(\Delta-3/2)^{5}\,\Gamma(\Delta+1/2)^{5/2}}\kappa^{5/2}(q-q_{c})^{5/2}\
,$ (236)

where

$\displaystyle q_{c}$
$\displaystyle=\frac{2^{2\Delta-2}\sqrt{3}\,\Gamma(\Delta-3/2)^{2}\,\Gamma(\Delta+1/2)}{5\kappa\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}\theta\
,$ (237) $\displaystyle m_{sol}$ $\displaystyle=\sqrt{3}\kappa
q+\left(\frac{5\,\Gamma(\Delta)\,\Gamma(2\Delta-5/2)}{2^{2\Delta}\,\Gamma(\Delta-3/2)^{2}\,\Gamma(\Delta+1/2)}\kappa^{2}q^{2}-\frac{\sqrt{3}}{2}\kappa
q\theta\right)\ .$ (238)

In $q<q_{c}$, no real solution satisfying $\mu_{RN}=\mu_{sol}$ was obtained.

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II | IY | IY | 6.0048e-06
II | ZZ | ZZ | 5.3697e-06
ZY | ZI | II | 5.2953e-06
II | ZX | IZ | 4.8373e-06
ZX | ZZ | II | 4.6782e-06
II | ZX | ZY | 4.6650e-06
YI | XI | II | 4.6247e-06
ZX | IY | II | 4.5411e-06
Average 1-location $p^{(2)}_{1}$ | 7.3897e-06
Average 2-location $p^{(2)}_{2}$ | 2.5013e-04
Average 3-location $p^{(2)}_{3}$ | 1.0883e-04
Table 6: Pauli errors with largest probabilities for simultaneous CNOT-gate
after optimization conducted in subsection IV.5. This is the Pauli errors
converted from the unitary error using Equation 9 and we have not added
decoherence error yet. The color of each row serves as a visual indicator of
the error’s weight.