††institutetext: School of Physics, Sun Yat-sen University, Guangzhou, 510275,
China

# Holographic supersymmetric Rényi entropies from hyperbolic black holes with
scalar hair

Jie Ren  and Dao-Quan Sun<EMAIL_ADDRESS>
###### Abstract

We study holographic supersymmetric Rényi entropies from a family of
hyperbolic black holes in an Einstein-Maxwell-dilaton (EMD) system under the
BPS condition. We calculate the thermodynamic quantities of these hyperbolic
black holes. We find a remarkably simple formula of the supersymmetric Rényi
entropy that unifies (interpolates) 11 cases embeddable to 10 or 11
dimensional supergravity. It reproduces many known results in the literature,
and gives new results with distinctive features. We show that the
supersymmetric version of the modular entropy and the capacity of entanglement
cannot be mapped to thermal quantities, due to the dependence of the
temperature and the chemical potential by the BPS condition. We also calculate
the entanglement spectrum. We derive the potential of the EMD system from a
$V=0$ solution and obtain two neutral solutions with scalar hair as a
byproduct.

##  1 Introduction

The Rényi entropy is a one-parameter generalization of the entanglement
entropy, quantifying the degree of entanglement between two subsystems of a
quantum system $A\cup B$:

$S_{n}=\frac{1}{1-n}\log\text{Tr}[\rho^{n}_{A}],$ (1)

where $\rho_{A}=\text{Tr}_{B}\rho_{AB}$ is the reduced density operator of
$A$. The Rényi entropy can be computed by the replica method with the path
integral on an $n$-fold cover of the original Euclidean geometry branched
around the entanglement surface $\partial A$ Calabrese:2004eu ;
Calabrese:2009qy . The supersymmetric extension of the Rényi entropy was
introduced in Nishioka:2013haa  and studied in various dimensions
Nishioka:2014mwa ; Crossley:2014oea ; Huang:2014gca ; Hama:2014iea ;
Alday:2014fsa ; Huang:2014pda ; Zhou:2015cpa ; Zhou:2015kaj ; Nian:2015xky ;
Giveon:2015cgs ; Mori:2015bro ; Nishioka:2016guu ; Yankielowicz:2017xkf ;
Hosseini:2019and . The supersymmetric Rényi entropy is defined as

$S^{\text{susy}}_{n}=\frac{1}{1-n}\text{log}\left|\frac{Z^{\text{susy}}_{n}}{(Z^{\text{susy}}_{1})^{n}}\right|,$
(2)

where $Z^{\text{susy}}_{n}$ is the supersymmetric partition function on a
$n$-branched manifold by turning on an $R$-symmetry background field to
preserve supersymmetries. An advantage of the supersymmetric Rényi entropy is
that it can be calculated using the localization technique Pestun:2007rz ;
Kapustin:2009kz ; Jafferis:2010un ; Hama:2010av ; Hama:2011ea . By taking the
strong coupling and large $N$ limit, it is matched to the holographic
calculations by gravity.

The AdS/CFT correspondence offers a convenient way to calculate the Rényi
entropy for CFTs in the ground state with a spherical entangling surface
Casini:2011kv ; Hung:2011nu . The ordinary Rényi entropy can be mapped to the
thermal entropy in the $\mathbb{S}^{1}\times\mathbb{H}^{d-1}$ with the
temperature $T=1/(2\pi Ln)$, where $L$ is the radius of the hyperbolic space
$\mathbb{H}^{d-1}$. In terms of thermal partition functions,

$Z[\mathbb{S}^{d}]=Z[\mathbb{S}^{1}\times\mathbb{H}^{d-1}].$ (3)

Similarly, the supersymmetric Rényi entropy can be mapped to a twisted thermal
partition function. The holographic dual to
$\mathbb{S}^{1}\times\mathbb{H}^{d-1}$ is a hyperbolic black hole (up to a
Wick rotation). The supersymmetric Rényi entropy can be calculated in terms of
charged hyperbolic black holes under the Bogomol’nyi-Prasad-Sommerfield (BPS)
condition. The charged Rényi entropy is a generalization of the Rényi entropy,
and it takes into account the distribution of a conserved charge across the
entangled states Belin:2013uta . While the holographic charged Rényi entropy
can be calculated by means of charged hyperbolic black holes, the BPS
condition imposes a constraint between the temperature and the chemical
potential of the black hole.

In this paper, we compute holographic supersymmetric Rényi entropy by means of
charged hyperbolic black holes in an Einstein-Maxwell-dilaton (EMD) system
that interpolates truncations of the following top-down models under the BPS
condition.

  * •

AdS5/CFT4: $U(1)^{3}$ truncations of $D=5$ gauged supergravity embeddable to
AdS${}_{5}\times\mathbb{S}^{5}$, near-horizon limit of rotating D3-brane in
10D type IIB supergravity. The CFT dual is $\mathcal{N}=4$ supersymmetric
Yang–Mills (SYM) theory Maldacena:1997re . Three special cases can be reduced
to an EMD system (including when the dilaton vanishes).

  * •

AdS4/CFT3: $U(1)^{4}$ truncations of $D=4$ gauged supergravity embeddable to
AdS${}_{4}\times\mathbb{S}^{7}$, near-horizon limit of rotating M2-brane in
11D supergravity. The CFT dual is the ABJM model Aharony:2008ug . Four special
cases can be reduced to an EMD system (including when the dilaton vanishes).

  * •

AdS7/CFT6: $U(1)^{2}$ truncations of $D=7$ gauged supergravity embeddable to
AdS${}_{7}\times\mathbb{S}^{4}$, near-horizon limit of rotating M5-brane in
11D supergravity. The CFT dual is 6D $(2,0)$ superconformal field theory. Two
special cases can be reduced to an EMD system.

  * •

AdS6/CFT5: Romans F(4) gauged supergravity Romans:1985tw  coupled to matter
embeddable to a warped AdS${}_{6}\times\mathbb{S}^{4}$ background of massive
IIA supergravity Intriligator:1997pq . Two special cases can be reduced to an
EMD system.

The metrics of these $D=4,5,6,7$ supergravity theories are summarized in
appendix A. Among these solutions, there are 11 cases of EMD truncations.
Interestingly, they can be interpolated by a single EMD system. When the
interpolating parameter $\alpha$ takes special values for $AdS_{d+1}$, the EMD
system coincides with top-down models. When $\alpha$ does not take these
special values, the EMD system in AdS4 still belongs to supergravity. Then we
can treat the EMD system in AdS4 as a bottom-up model, since the supersymmetry
is gauged in the bulk and global in the boundary.

We find a remarkably simple formula of the supersymmetric Rényi entropy that
unifies (interpolates) the above 11 special cases; see (30) below. Among these
11 cases of the supersymmetric Rényi entropy, many have appeared in the
literature in SCFT and gravity calculations, and some have not appeared in the
literature and have new features.

We also calculate the capacity of entanglement PhysRevLett.105.080501  and the
entanglement spectrum Haldane:2008en , which are entanglement data expressed
in different ways than the Rényi entropy. The capacity of entanglement as a
quantum information measure different from entanglement entropy has been
studied broadly DeBoer:2018kvc ; Nakaguchi:2016zqi ; Nakagawa:2017wis ;
Sun:2023tmv . In this paper, we generalize the capacity of entanglement to the
supersymmetric capacity of entanglement and show that it cannot be mapped to
the standard heat capacity of the thermal CFT on hyperbolic space, due to the
dependence of the temperature and the chemical potential by the BPS condition.
This is different from the previous results of the non-supersymmetric capacity
of entanglement. We also calculate the entanglement spectrum, which are
eigenvalues of $\rho_{A}$. We write the entanglement spectrum as convolutions
of generalized hypergeometric functions.

At first glance, the dilaton potential in our model looks a little cumbersome.
However, we demonstrate that this is the most natural way to add a
cosmological constant to the Horowitz-Strominger solution Horowitz:1991cd
found in 1991. We derive the potential of the EMD system starting with the
$V=0$ solution under reasonable assumptions. As a byproduct, we obtain two
nontrivial neutral limits as hyperbolic black holes with scalar hair.

The paper is organized as follows. In section 2, we take advantage of an EMD
system to calculate the supersymmetric Rényi entropy, and obtain a simple
result that interpolates 11 special cases from top-down truncations. In
section 3, we study the capacity of entanglement and the entanglement
spectrum. In section 4, we derive the potential of the scalar field. In
section 5, we conclude with some open questions. In appendix A, we review
consistent truncations of $D=10$ and $D=11$ solutions in supergravity. In
appendix B, we review the FI-gauged supergravity. In appendix C, we give a
special IR geometry.

##  2 Supersymmetric Rényi entropy from EMD systems

We start with an EMD system that has large intersections with supergravities.
We will give an elegant explanation in section 4.1 that the dilaton potential
$V(\phi)$ is naturally generated from the $V=0$ solution. The
$(d+1)$-dimensional action is

$S=\int
d^{d+1}x\sqrt{-g}\biggl{(}R-\frac{1}{4}e^{-\alpha\phi}F^{2}-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\biggr{)},$
(4)

where $d\geq 3$, $F=dA$, and $\alpha$ is a parameter. We have set $16\pi G=1$.
The potential of the dilaton field is

$V(\phi)=v_{1}e^{-\frac{2(d-2)}{(d-1)\alpha}\phi}+v_{2}e^{\frac{(d-1)\alpha^{2}-2(d-2)}{2(d-1)\alpha}\phi}+v_{3}e^{\alpha\phi}\,,$
(5)

where

$\begin{split}&v_{1}=-\frac{(d-1)^{2}[d(d-1)\alpha^{2}-2(d-2)^{2}]\alpha^{2}}{[2(d-2)+(d-1)\alpha^{2}]^{2}L^{2}}\,,\\\
&v_{2}=-\frac{8(d-1)^{3}(d-2)\alpha^{2}}{{[2(d-2)+(d-1)\alpha^{2}]^{2}L^{2}}}\,,\\\
&v_{3}=-\frac{2(d-1)(d-2)^{2}[2d-(d-1)\alpha^{2}]}{[2(d-2)+(d-1)\alpha^{2}]^{2}L^{2}}\,.\end{split}$
(6)

Table 1 below clearly justifies this potential by showing that it reproduces
11 EMD systems of especially physical significance. The potential can be
written in terms of a superpotential $W(\phi)$ Lu:2013eoa :

$\displaystyle V(\phi)$
$\displaystyle=W^{\prime}(\phi)^{2}-\frac{d}{2(d-1)}W^{2},$ (7) $\displaystyle
W(\phi)$
$\displaystyle=\frac{2\sqrt{2}\,(d-1)(d-2)}{[2(d-2)+(d-1)\alpha^{2}]L}\biggl{(}e^{\frac{\alpha}{2}\phi}+\frac{(d-1)\alpha^{2}}{2(d-2)}e^{-\frac{d-2}{(d-1)\alpha}\phi}\biggr{)}.$
(8)

The $\phi\to 0$ behavior of $V(\phi)$ is

$V(\phi)=-\frac{d(d-1)}{L^{2}}-\frac{d-2}{L^{2}}\phi^{2}+\mathcal{O}(\phi^{3})\,,$
(9)

where the first term is the cosmological constant, and the second term shows
that the mass of the scalar field satisfies $m^{2}L^{2}=-2(d-2)$. The scaling
dimension of the scalar operator dual to the scalar field $\phi$ satisfies
$\Delta(\Delta-d)=m^{2}L^{2}$, which has two solutions $\Delta_{\pm}=2$,
$d-2$. Recall that the alternative quantization exists when $-d^{2}/4\leq
m^{2}\leq-d^{2}/4+1$. The mass saturates the BF bound in $d=4$; the
alternative quantization exists only in $d=3$.

The above EMD system admits an analytic solution Gao:2004tu ; Gao:2004tv ;
Gao:2005xv :

$\displaystyle ds^{2}=-f(r)\hskip 0.83328ptdt^{2}+\frac{1}{g(r)}\hskip
0.83328ptdr^{2}+U(r)\hskip 0.83328ptd\Sigma_{d-1,\hskip 0.5833ptk}^{2}\,,$
(10) $\displaystyle A=2\sqrt{\frac{(d-1)\hskip 0.83328ptb\hskip
0.83328ptc}{2(d-2)+(d-1)\alpha^{2}}}\biggl{(}\frac{1}{r_{h}^{d-2}}-\frac{1}{r^{d-2}}\biggr{)}dt\,,$
(11) $\displaystyle
e^{\alpha\phi}=\biggl{(}1-\frac{b}{r^{d-2}}\biggr{)}^{\frac{2(d-1)\alpha^{2}}{2(d-2)+(d-1)\alpha^{2}}},$
(12)

where $d\Sigma_{d-1,\hskip 0.5833ptk}^{2}$ is a $(d-1)$-dimensional hyperbolic
space $\mathbb{H}^{d-1}$ ($k=-1$), plane $\mathbb{R}^{d-1}$ ($k=0$), or sphere
$\mathbb{S}^{d-1}$ ($k=1$) of unit radius. The functions in the metric are

$\displaystyle f$
$\displaystyle=\biggl{(}k-\frac{c}{r^{d-2}}\biggr{)}\biggl{(}1-\frac{b}{r^{d-2}}\biggr{)}^{\frac{2(d-2)-(d-1)\alpha^{2}}{2(d-2)+(d-1)\alpha^{2}}}+\frac{r^{2}}{L^{2}}\biggl{(}1-\frac{b}{r^{d-2}}\biggr{)}^{\frac{2(d-1)\alpha^{2}}{(d-2)[2(d-2)+(d-1)\alpha^{2}]}},$
$\displaystyle g$
$\displaystyle=f(r)\biggl{(}1-\frac{b}{r^{d-2}}\biggr{)}^{\frac{2(d-3)(d-1)\alpha^{2}}{(d-2)[2(d-2)+(d-1)\alpha^{2}]}},$
(13) $\displaystyle U$
$\displaystyle=r^{2}\biggl{(}1-\frac{b}{r^{d-2}}\biggr{)}^{\frac{2(d-1)\alpha^{2}}{(d-2)[2(d-2)+(d-1)\alpha^{2}]}}.$

The system is invariant under $\alpha\to-\alpha$ and $\phi\to-\phi$; we assume
$\alpha\geq 0$. When $\alpha=0$, we obtain the RN-AdSd+1 black hole. Later we
take $k=-1$ for hyperbolic black holes. We set $L=1$ in the following.

The mass, temperature, entropy, chemical potential, and charge are given by
Sheykhi:2009pf ; Hendi:2010gq 111For a derivation of the mass by holographic
renormalization in AdS4, see appendix A of Bai:2022obp .

$\displaystyle M=\frac{(d-1)V_{\Sigma}}{16\pi
G}\biggl{(}c+k\,\frac{2(d-2)-(d-1)\alpha^{2}}{2(d-2)+(d-1)\alpha^{2}}\,b\biggr{)},$
(14) $\displaystyle
T=\frac{\sqrt{f^{\prime}g^{\prime}}}{4\pi}\biggr{|}_{r=r_{h}},\qquad
S=\frac{V_{\Sigma}}{4G}U(r_{h})^{(d-1)/2}\,,$ (15)
$\displaystyle\mu=2\sqrt{\frac{(d-1)\hskip 0.83328ptb\hskip
0.83328ptc}{2(d-2)+(d-1)\alpha^{2}}}\frac{1}{r_{h}^{d-2}}\,,\qquad
Q=2(d-2)V_{\Sigma}\sqrt{\frac{(d-1)\hskip 0.83328ptb\hskip
0.83328ptc}{2(d-2)+(d-1)\alpha^{2}}}\,,$ (16)

where $V_{\Sigma}$ is the volume of $\mathbb{H}^{d-1}$, regulated by
integrating out to a maximum radius $R$ of this hyperbolic space Hung:2011nu :

$V_{\Sigma}\simeq\frac{\Omega_{d-2}}{d-2}\biggl{[}\frac{R^{d-2}}{\delta^{d-2}}-\frac{(d-2)(d-3)}{2(d-4)}\frac{R^{d-4}}{\delta^{d-4}}+\cdots\biggr{]},$
(17)

where $\Omega_{d-2}=2\pi^{(d-1)/2}/\Gamma((d-2)/2)$ is the area of
$\mathbb{S}^{d-2}$. The cutoff $\delta$ is related to the UV cutoff in the
dual CFT, consistent with the area law of the entanglement entropy. We have
checked that the first law of thermodynamics $dM=TdS+\mu dQ$ is satisfied by
(14)–(16). In the grand canonical ensemble, we use the grand potential
$\Omega=M-TS-\mu Q$, by which the first law of thermodynamics is

$d\Omega=-SdT-Qd\mu\,.$ (18)

The heat capacity at fixed chemical potential is

$C_{\mu}=T\left(\frac{\partial S}{\partial
T}\right)_{\mu}=-T\frac{\partial^{2}\Omega(T,\mu)}{\partial
T^{2}}\bigg{|}_{\mu}.$ (19)

In the following, we focus on the hyperbolic black holes under the “BPS
condition”

$c=-b\,.$ (20)

Recall that the BPS condition for a solution in supergravity is the condition
that the Killing spinor equation has nontrivial solutions.222The technical
details to solve the Killing spinor equation can be found in Romans:1991nq ,
for example. When it is satisfied, the black hole is supersymmetric, and the
corresponding state in the CFT is supersymmetric. This EMD system is related
to consistent truncations of supergravity in the following way.

  * •

When the parameters $\alpha$ and $d$ take the values in table 1, we have
checked that (20) is exactly the BPS condition for all 11 cases according to
previous studies on $D=4$, $5$, $6$, $7$ supergravity solutions Huang:2014pda
; Zhou:2015kaj ; Hosseini:2019and  as consistent truncations of $D=10$ and
$D=11$ supergravities. In these cases, the EMD system is a top-down model.

  * •

For AdS4, the system belongs to supergravity for all $\alpha$. The EMD system
is obtained by turning off one of two U(1) gauge fields in an FI-gauged
supergravity; see appendix B. Here (20) is the BPS condition according to
Nozawa:2022upa . The system is a bottom-up model that interpolates four top-
down models.

  * •

For AdS5 and higher dimensions, the system may belong to the so-called fake
supergravity Freedman:2003ax ; Celi:2004st , in which we treat (20) as a BPS-
like condition.

We need to distinguish the extremal limit and the BPS limit for hyperbolic
black holes in AdS. For the (asymptotically flat) RN black hole, these two
limits coincide. However, in the AdS case, they are not the same.

  * •

The extremal limit of a black hole is reached when (i) two horizons merge to a
degenerate horizon, or (ii) the horizon moves towards the spacetime
singularity. In case (i) we obtain an AdS2 factor in the extremal limit. In
case (ii) we may obtain hyperscaling-violating geometries.

  * •

The BPS limit is different from the extremal limit for hyperbolic black holes
in AdS. By taking the BPS limit, we can still vary the temperature. For a
certain range of $\alpha$, the extremal limit can be taken, and the
temperature reaches zero.

For the solution (10)–(13), the gauge field is imaginary under the BPS
condition $c=-b$. Nevertheless, all thermodynamical quantities are well
defined, and the gauge field is real in the Euclidean signature, the same as
for the hyperbolic RN-AdS black hole Belin:2013uta . The curvature singularity
is at $r=0$ and $r^{d-2}=b$, and the parameter $b$ can be either positive or
negative. The horizon of the black hole is determined by $f(r_{h})=0$, from
which the parameter $b$ is expressed in terms of $r_{h}$:

$b=r_{h}^{d-2}-r_{h}^{\frac{(d-2)^{2}[2d-(d-1)\alpha^{2}]}{2(d-2)^{2}-d(d-1)\alpha^{2}}}.$
(21)

The temperature is given by

$T=\frac{2(d-1)(d-2)r_{h}^{p}-2(d-2)^{2}+(d-1)\alpha^{2}}{2\pi[2(d-2)+(d-1)\alpha^{2}]},$
(22)

where we have used (21) to replace $b$ with $r_{h}$, and

$p=\frac{(d-2)[2(d-2)+(d-1)\alpha^{2}]}{2(d-2)^{2}-(d-1)\alpha^{2}}.$ (23)

From (22), we conclude that the temperature can reach zero only if
$0\leq\alpha\leq\alpha_{*}$, where

$\alpha_{*}=(d-2)\sqrt{\frac{2}{d-1}}\,.$ (24)

There are three distinctive classes as follows:

  * •

$0\leq\alpha<\alpha_{*}$. The temperature reaches zero when

$r_{h}=\biggl{(}\frac{d-2}{d-1}-\frac{\alpha^{2}}{2(d-2)}\biggr{)}^{\frac{1}{2p}}.$
(25)

At zero temperature, the IR geometry is AdS${}_{2}\times\mathbb{H}^{d-1}$.
Notice that the $\alpha=0$ case is the RN-AdSd+1 black hole.

  * •

$\alpha=\alpha_{*}$. The horizon size is $r_{h}=L$. The temperature is
$T=\sqrt{2(1-b)}/4\pi$. The temperature reaches zero when $b=1$. At zero
temperature, the IR geometry has a curvature singularity. See appendix C for
the IR geometry.

  * •

$\alpha>\alpha_{*}$. The temperature cannot reach zero. There is a minimal
temperature at $r_{h}=0$.

After a conformal mapping, the Rényi entropy with the entangling surface being
a sphere can be calculated from the thermodynamics of the CFT living on
$\mathbb{S}\times\mathbb{H}^{d-1}$ Casini:2011kv ; Hung:2011nu . By the
AdS/CFT correspondence, the supersymmetric Rényi entropy can be calculated in
terms of the hyperbolic black holes under the BPS condition. We have

$\text{Tr}[\rho_{A}^{n}]=\frac{Z(T_{0}/n)}{Z(T_{0})^{n}},$ (26)

where $Z(T)=\text{Tr}[e^{H/T}]$ is the thermal partition function of the
hyperbolic black hole at temperature $T$. For the entangling surface with
radius $L$, we have $T_{0}=1/(2\pi L)$ and $T=T_{0}/n$. With the grand
potential $\Omega=\beta^{-1}I=-T\log Z$ of black holes, where $I$ is the
Euclidean on-shell action and $\beta=1/T$. The supersymmetric Rényi entropy is
given by

$S_{n}=\frac{n}{1-n}\frac{1}{T_{0}}\bigl{[}\Omega(T_{0},\mu_{0})-\Omega(T_{0}/n,\mu)\bigr{]}.$
(27)

The integral representation of $S_{n}$ is Huang:2014gca

$S_{n}=\frac{n}{n-1}\int_{n}^{1}\partial_{n^{\prime}}\left(\frac{\log
Z(T_{0}/n^{\prime},\mu)}{n^{\prime}}\right)dn^{\prime}=\frac{n}{n-1}\frac{1}{T_{0}}\int_{T_{0}/n}^{T_{0}}\left(S+Q\frac{d\mu}{dT}\right)dT.$
(28)

Note that $T$ and $\mu$ are not independent due to BPS condition.

For our hyperbolic black holes, the grand potential under the condition $c=-b$
is

$\Omega=M-TS-\mu Q=-\frac{V_{\Sigma}}{16\pi
G}r_{h}^{\frac{2(d-1)(d-2)^{2}}{2(d-2)^{2}-(d-1)\alpha^{2}}}\,.$ (29)

The supersymmetric Rényi entropies are given by

$S_{n}=\frac{V_{\Sigma}}{4G}\,\frac{n}{n-1}\Biggl{[}1-\left(\frac{(d-2)n+1}{(d-1)n}-\frac{(n-1)\alpha^{2}}{2(d-2)n}\right)^{\frac{2(d-1)(d-2)}{2(d-2)+(d-1)\alpha^{2}}}\Biggr{]}.$
(30)

The entanglement entropy is

$S_{1}=\lim_{n\to 1}S_{n}=\frac{V_{\Sigma}}{4G}\quad(\,=4\pi V_{\Sigma}\,)\,,$
(31)

which is the same for all $\alpha$. To have a well-defined $S_{n}$ for all
$n$, we need $\alpha\leq\alpha_{*}$ For $d=3,4,5,6$, the values of
$\alpha_{*}$ are as follows.

| AdS4 | AdS5 | AdS6 | AdS7
---|---|---|---|---
$\alpha_{*}$ | 1 | $4/\sqrt{6}$ | $3/\sqrt{2}$ | $8/\sqrt{10}$
(32)

As we will see in the next subsection, an inequality of the Rényi entropy is
always violated when $\alpha>\alpha_{*}$.

Table 1:  “Periodic table” of top-down supergravity models and special cases of the supersymmetric Rényi entropy. The naming convention and the metrics are in appendix A. (A) The $S_{n}$ was calculated by the localization method in the SCFT and matches the holographic result. (B) The $S_{n}$ was calculated for free fields, and matches the holographic result. (C) To our knowledge, no SCFT calculation is known. (D) It violates Rényi entropic inequalities. AdSd+1 | $\alpha$ | Name | Metric | Supersymmetric Rényi entropy | Cases
---|---|---|---|---|---
AdS4 ($d=3$) | $\alpha=0$ |  RN-AdS4 | $H_{1,2,3,4}=H$ | $S_{n}=\dfrac{3n+1}{4n}S_{1}$ |  A Nishioka:2013haa ; Huang:2014gca ; Nishioka:2014mwa
$\alpha=1/\sqrt{3}$ | 3-charge |  $H_{1,2,3}=H$, $H_{4}=1$ | $S_{n}=\dfrac{n}{n-1}\biggl{[}1-\biggl{(}\dfrac{n+2}{3n}\biggr{)}^{3/2}\biggr{]}S_{1}$ | C
$\alpha=1$ | 2-charge |  $H_{1,2}=H$, $H_{3,4}=1$ | $S_{n}=S_{1}$ | C
$\alpha=\sqrt{3}$ | 1-charge |  $H_{1}=H$, $H_{2,3,4}=1$ | $S_{n}=\dfrac{n}{n-1}\biggl{[}1-\biggl{(}\dfrac{2-n}{n}\biggr{)}^{1/2}\biggr{]}S_{1}$ | C, D
AdS5 ($d=4$) | $\alpha=0$ |  RN-AdS5 | $H_{1,2,3}=H$ | $S_{n}=\dfrac{19n^{2}+7n+1}{27n^{2}}S_{1}$ |  B Huang:2014pda ; Zhou:2015cpa
$\alpha=2/\sqrt{6}$ | 2-charge |  $H_{1,2}=H$, $H_{3}=1$ | $S_{n}=\dfrac{3n+1}{4n}S_{1}$ |  A, B Huang:2014pda
$\alpha=4/\sqrt{6}$ | 1-charge |  $H_{1}=H$, $H_{2,3}=1$ | $S_{n}=S_{1}$ |  B Huang:2014pda
AdS6 ($d=5$) | $\alpha=1/\sqrt{2}$ | 2-charge | $H_{1}=H_{2}=H$ | $S_{n}=\dfrac{19n^{2}+7n+1}{27n^{2}}S_{1}$ |  A Hosseini:2019and ; Hama:2014iea ; Alday:2014fsa
$\alpha=5/\sqrt{10}$ | 1-charge |  $H_{1}=H$, $H_{2}=1$ | $S_{n}=\dfrac{n}{n-1}\biggl{[}1-\biggl{(}\dfrac{n+2}{3n}\biggr{)}^{3/2}\biggr{]}S_{1}$ | C
AdS7 ($d=6$) | $\alpha=2/\sqrt{10}$ | 2-charge | $H_{1}=H_{2}=H$ | $S_{n}=\dfrac{175n^{3}+67n^{2}+13n+1}{256n}S_{1}$ |  B Zhou:2015kaj
$\alpha=6/\sqrt{15}$ | 1-charge |  $H_{1}=H$, $H_{2}=1$ | $S_{n}=\dfrac{3n+1}{4n}S_{1}$ |  B Zhou:2015kaj

For 11 special cases belonging to top-down models, we list them with their
supersymmetric Rényi entropies in table 1. For AdS4 and AdS5, $\alpha=0$ gives
the RN black hole. We observe distinctive features as follows. The Rényi
entropy reproduces known results (A) and (B) while giving new features (C) and
(D).

  * (A)

RN-AdS4, 2-charge black hole in AdS5, 2-charge black hole in AdS6: The SCFT
calculations of $S_{n}$ have been performed by the localization technique and
they match the holographic result Nishioka:2013haa ; Huang:2014gca ;
Nishioka:2014mwa ; Huang:2014pda ; Hosseini:2019and ; Hama:2014iea ;
Alday:2014fsa .

  * (B)

All cases in AdS5 and AdS7: The $S_{n}$ was calculated by the heat kernel
method for free fields, and they match the holographic result Huang:2014pda ;
Zhou:2015cpa ; Zhou:2015kaj .

  * (C)

1-, 2-, and 3-charge black holes in AdS4 and 1-charge black hole in AdS6:
These cases that are not compared with SCFT calculations. The holographic
result can be obtained by special cases of Hosseini:2019and .333In a private
communication, we learned that Yang Zhou has obtained the $S_{n}$ for 1, 2,
and 3-charge black holes in AdS4, though these findings were not published.

  * (D)

1-charge black hole in AdS4 is peculiar. When $d=3$, $\alpha=\sqrt{3}$, we
have $r_{h}=\sqrt{n/(2-n)}$, which is real only when $0<n<2$. This is the only
case in which $\alpha>\alpha_{*}$ among the 11 cases.

As a remark, the BPS condition (20) significantly simplifies the
thermodynamical quantities. If the parameters $b$ and $c$ are arbitrary, no
explicit solution is available for the Rényi entropy $S_{n}$. Another
condition in which $S_{n}$ is explicitly solvable is $c=0$, which was studied
in detail in Bai:2022obp , based on a nontrivial neutral limit of these black
holes Ren:2019lgw .

##  3 Exporing the entanglement data

###  3.1 Modular entropy and capacity of entanglement

The Rényi entropy as an information-theoretic quantity is related to the
thermal entropy on $\mathbb{S}^{1}\times\mathbb{H}^{d-1}$. The inequalities
for the Rényi entropy have been proposed in quantum information Beck:1993  and
studied holographically Hung:2011nu :

$\displaystyle\frac{\partial}{\partial n}\left(\frac{n-1}{n}S_{n}\right)$
$\displaystyle\geq 0,$ (33) $\displaystyle\frac{\partial^{2}}{\partial
n^{2}}\left((n-1)S_{n}\right)$ $\displaystyle\leq 0.$ (34)

The first one corresponds to the positivity of the modular entropy, and the
second one corresponds to the positivity of the specific heat. The modular
entropy has a geometric interpretation Dong:2016fnf . The capacity of
entanglement as an important measure of quantum information was originally
introduced in PhysRevLett.105.080501 , and then studied in holography
Nakaguchi:2016zqi . For the supersymmetric Rényi entropy calculated by
hyperbolic holes, we find that these inequalities are not satisfied when
$\alpha>\alpha_{*}$.

In terms of the supersymmetric Rényi entropy, we define the supersymmetric
modular entropy as

$\widetilde{S}^{\text{susy}}_{n}=n^{2}\partial_{n}\Bigl{(}\frac{n-1}{n}S_{n}^{\text{susy}}\Bigr{)},$
(35)

and the supersymmetric capacity of entanglement as

$C_{E}^{\text{susy}}(n)=n^{2}\partial_{n}^{2}[(1-n)S_{n}^{\text{susy}}].$ (36)

When $n\to 1$,
$C_{E}^{\text{susy}}(1)=\langle(-\text{log}\rho_{A})^{2}\rangle-\langle-\text{log}\rho_{A}\rangle^{2}$
gives the quantum fluctuation with respect to the original state $\rho_{A}$.

We find that the modular entropy $\widetilde{S}_{n}^{\text{susy}}$ no longer
equals the thermal entropy of the hyperbolic black hole due to the fact that
the BPS condition puts a constraint on the temperature and the chemical
potential. Similarly, the capacity of entanglement $C_{E}^{\text{susy}}(n)$
cannot map to the heat capacity $C_{\mu}$ of the thermal CFT on hyperbolic
space, unlike the non-supersymmetric capacity of entanglement. More precisely,

$\displaystyle\widetilde{S}_{n}^{\text{susy}}$
$\displaystyle=S+Q\frac{d\mu}{dT}$ (37) $\displaystyle C_{E}^{\text{susy}}(n)$
$\displaystyle=C_{\mu}+3T\hskip 0.83328pt\frac{dQ}{dT}\hskip
0.83328pt\frac{d\mu}{dT}+TQ\hskip 0.83328pt\frac{d^{2}\mu}{dT^{2}}.$ (38)

The latter was obtained as follows. From (27) and (36), by $T=T_{0}/n$, we
obtain

$\displaystyle C_{E}^{\text{susy}}(n)$
$\displaystyle=-\frac{d}{dT}\biggl{[}T^{2}\frac{d}{dT}\biggl{(}\frac{1}{T}\Omega(T,\mu)\biggr{)}\biggr{]}=-T\frac{d^{2}}{dT^{2}}\Omega(T,\mu)$
(39) $\displaystyle=C_{\mu}-2T\hskip
0.83328pt\frac{\partial^{2}\Omega(T,\mu)}{\partial T\partial\mu}\hskip
0.83328pt\frac{d\mu}{dT}-T\hskip
0.83328pt\frac{\partial^{2}\Omega(T,\mu)}{\partial\mu^{2}}\biggl{(}\frac{d\mu}{dT}\biggr{)}^{2}-T\hskip
0.83328pt\frac{\partial\Omega(T,\mu)}{\partial\mu}\hskip
0.83328pt\frac{d^{2}\mu}{dT^{2}}.$

where the chemical potential $\mu$ and the charge $Q$ depend on the
temperature $T$ for supersymmetric theories.

From (30) and (36), we obtain the capacity of entanglement

$C_{E}(n)=\frac{1}{n}\left(\frac{(d-2)n+1}{(d-1)n}-\frac{(n-1)\alpha^{2}}{2(d-2)n}\right)^{\frac{2(d-1)(d-2)}{2(d-2)+(d-1)\alpha^{2}}-2}C_{E}(1),$
(40)

where

$C_{E}(1)=\biggl{(}\frac{d-2}{d-1}-\frac{\alpha^{2}}{2(d-2)}\biggr{)}S_{1}.$
(41)

For 11 special cases belonging to top-down models, we list their
supersymmetric capacity of entanglement in table 2. The ratio $C_{E}(1)/S_{1}$
of the capacity of entanglement yields universal information characterizing
the dual CFTs DeBoer:2018kvc . From above result, we can see that the ratio
$C_{E}(1)/S_{1}$ of the supersymmetric capacity of entanglement is different
from the ratio $C_{E}(1)/S_{1}=1$ of the non-supersymmetric capacity of
entanglement for neutral black holes DeBoer:2018kvc ; Sun:2023tmv .

We can define a heat capacity for black holes under the BPS condition

$C_{\text{BPS}}=T\frac{dS}{dT}=\frac{1}{n}\left(\frac{(d-2)n+1}{(d-1)n}-\frac{(n-1)\alpha^{2}}{2(d-2)n}\right)^{\frac{2(d-1)(d-2)}{2(d-2)+(d-1)\alpha^{2}}-1}S_{1}\,.$
(42)

It can be verified that the above result equals $(1/n)\widetilde{S}_{n}$,
where $\widetilde{S}_{n}$ is calculated by (35).

Table 2:  Special cases of the supersymmetric entanglement of capacity. AdSd+1 | $\alpha$ | Name | $S_{n}$ |  $\quad C_{E}(t)\quad(\hskip 0.66663ptt\equiv 1/n=2\pi T$ )
---|---|---|---|---
AdS4 ($d=3$) | $\alpha=0$ |  RN-AdS4 | $\dfrac{3n+1}{4n}S_{1}$ | $C_{E}(t)=\dfrac{1}{2}tS_{1}$
$\alpha=1/\sqrt{3}$ | 3-charge | $S_{n}=\dfrac{n}{n-1}\biggl{[}1-\biggl{(}\dfrac{n+2}{3n}\biggr{)}^{3/2}\biggr{]}S_{1}$ | $C_{E}(t)=t(3+6t)^{-1/2}S_{1}$
$\alpha=1$ | 2-charge | $S_{n}=S_{1}$ | $C_{E}(t)=0$
$\alpha=\sqrt{3}$ | 1-charge | $S_{n}=\dfrac{n}{n-1}\biggl{[}1-\biggl{(}\dfrac{2-n}{n}\biggr{)}^{1/2}\biggr{]}S_{1}$ | $C_{E}(t)=-t(-1+2t)^{-3/2}S_{1}$
AdS5 ($d=4$) | $\alpha=0$ |  RN-AdS5 | $S_{n}=\dfrac{19n^{2}+7n+1}{27n^{2}}S_{1}$ | $C_{E}(t)=\dfrac{2}{9}t(2+t)S_{1}$
$\alpha=2/\sqrt{6}$ | 2-charge | $S_{n}=\dfrac{3n+1}{4n}S_{1}$ | $C_{E}(t)=\dfrac{1}{2}tS_{1}$
$\alpha=4/\sqrt{6}$ | 1-charge | $S_{n}=S_{1}$ | $C_{E}(t)=0$
AdS6 ($d=5$) | $\alpha=1/\sqrt{2}$ | 2-charge | $S_{n}=\dfrac{19n^{2}+7n+1}{27n^{2}}S_{1}$ | $C_{E}(t)=\dfrac{2}{9}t(2+t)S_{1}$
$\alpha=5/\sqrt{10}$ | 1-charge | $S_{n}=\dfrac{n}{n-1}\biggl{[}1-\biggl{(}\dfrac{n+2}{3n}\biggr{)}^{3/2}\biggr{]}S_{1}$ | $C_{E}(t)=t(3+6t)^{-1/2}S_{1}$
AdS7 ($d=6$) | $\alpha=2/\sqrt{10}$ | 2-charge | $S_{n}=\dfrac{175n^{3}+67n^{2}+13n+1}{256n}S_{1}$ | $C_{E}(t)=\dfrac{3}{64}t(3+t)^{2}S_{1}$
$\alpha=6/\sqrt{15}$ | 1-charge | $S_{n}=\dfrac{3n+1}{4n}S_{1}$ | $C_{E}(t)=\dfrac{1}{2}tS_{1}$

###  3.2 Entanglement spectrum

The Rényi entropy $S_{n}$ for all $n$ determines the entanglement spectrum,
which is the eigenvalue distribution of the reduced density matrix $\rho_{A}$.
The holographic result of $S_{n}$ obtained in the last section is analytic at
$n=\infty$. Assuming this analyticity, the entanglement spectrum must include
both discrete and continuous parts, with one discrete eigenvalue $\lambda_{1}$
being the largest eigenvalue of the continuous spectrum Hung:2011nu . Thus,
the Rényi entropy can be written as

$S_{n}=\frac{1}{1-n}\log\text{Tr}[\rho^{n}]=\frac{1}{1-n}\log\biggl{[}d_{1}\lambda_{1}^{n}+\int_{0}^{\lambda_{1}}{\bar{\rho}(\lambda)\lambda^{n}d\lambda}\biggr{]},$
(43)

where $\bar{\rho}(\lambda)$ is the continuous part of the entanglement
spectrum $\rho(\lambda)$. By writing the discrete part into $\rho(\lambda)$
via a Dirac delta function, the Rényi entropy satisfies

$e^{(1-n)S_{n}}=\int_{t_{1}}^{+\infty}{e^{-(n+1)t}\rho(e^{-{t}})dt},$ (44)

where $\lambda$ is reparameterized as $\lambda=e^{-t}$, and
$\lambda_{1}=e^{-t_{1}}$. This is essentially a Laplace transform with $n$
being the parameter. Thus, the spectrum can be obtained from an inverse
Laplace transform,

$\rho(\lambda)=\frac{1}{\lambda}\mathcal{L}^{-1}\bigl{[}e^{(1-n)S_{n}},n,t\bigr{]}\bigl{|}_{t=-\log\lambda}=\frac{1}{\lambda}\frac{1}{2\pi
i}\lim_{T\rightarrow\infty}\int_{\gamma-
iT}^{\gamma+iT}{e^{(1-n)S_{n}}e^{nt}dn},$ (45)

where the integral is taken over a vertical line with $\text{Re}(s)=\gamma$,
and $\gamma$ is a real number ensuring no singularity on the right side of
this line.

Assuming that the Rényi entropy can be expanded near $n=\infty$ as

$S_{n}=\sum_{i=0}^{\infty}s_{i}n^{-i}=s_{0}+\frac{s_{1}}{n}+\frac{s_{2}}{n^{2}}+\cdots,$
(46)

where the constant term is related to the largest eigenvalue of the spectrum
by $\lambda_{1}=\exp(-s_{0})$. For the Rényi entropy given by (30), the
coefficients of the first two terms are

$\displaystyle\begin{split}s_{0}&=\frac{V_{\Sigma}}{4G}\Biggl{[}1-\left(\frac{d-2}{d-1}-\frac{\alpha^{2}}{2(d-2)}\right)^{\frac{2(d-1)(d-2)}{2(d-2)+(d-1)\alpha^{2}}}\Biggr{]},\\\
s_{1}&=s_{0}-\frac{V_{\Sigma}}{4G}\,\frac{2(d-1)(d-2)}{2(d-2)+(d-1)\alpha^{2}}\left(\frac{d-2}{d-1}-\frac{\alpha^{2}}{2(d-2)}\right)^{\frac{2(d-1)(d-2)}{2(d-2)+(d-1)\alpha^{2}}}.\end{split}$
(47)

From table 1, we can see that the series terminates at finite orders of
$n^{-1}$ in many cases.

In the following, we will give a way to express the entanglement spectrum,
which is the inverse Laplace transform of

$e^{(1-n)S_{n}}=e^{s_{0}-s_{1}}e^{-s_{0}n}\exp\biggl{(}\sum\limits_{i=1}^{\infty}{u_{i}n^{-i}}\biggr{)},\qquad
u_{i}=s_{i}-s_{i+1}.$ (48)

By the convolution theorem, we obtain

$\rho(\lambda)=\frac{e^{s_{0}-s_{1}}}{\lambda}\mathcal{L}^{-1}[e^{-s_{0}n}]*\mathcal{L}^{-1}[e^{u_{1}n^{-1}}]*\mathcal{L}^{-1}[e^{u_{2}n^{-2}}]*\cdots$
(49)

The convolution satisfies commutativity and associativity. The inverse Laplace
transform can be done term by term. The results are

$\displaystyle\frac{1}{\lambda}\hskip 0.83328pt\mathcal{L}^{-1}[e^{-s_{0}n}]$
$\displaystyle=\frac{1}{\lambda}\hskip
0.83328pt\delta(t-s_{0})=\delta(\lambda-\lambda_{1}),$ (50)
$\displaystyle\mathcal{L}^{-1}[e^{-s_{i}n^{-i}}]$
$\displaystyle=\delta(t)+\frac{u_{i}t^{i-1}}{(i-1)!}\,{{}_{0}F_{i}}\biggl{(};1+\frac{1}{i},1+\frac{2}{i},\cdots,2;\frac{u_{i}t^{i}}{i^{i}}\biggr{)}.$
(51)

where ${{}_{0}F_{i}}$ is a generalized hypergeometric function. Generally, it
is unlikely to write a closed form for the above convolutions. If the
large-$n$ expansion of $S_{n}$ terminates as $S_{n}=s_{0}+s_{1}/n$, which
happens three times in table 1, a closed form expression can be obtained in
the following. The function ${}_{0}F_{1}$ can be expressed in terms of the
modified Bessel function of the first kind:

${}_{0}F_{1}(;2;z)=\frac{1}{\sqrt{z}}I_{1}(2\sqrt{z})\,.$ (52)

As a consequence, the entanglement spectrum for $S_{n}=s_{0}+s_{1}/n$ is

$\rho(\lambda)=e^{s_{0}-s_{1}}\delta(\lambda_{1}-\lambda)+\frac{s_{1}\theta(\lambda_{1}-\lambda)}{\lambda\sqrt{s_{1}\ln(\lambda_{1}/\lambda)}}I_{1}\bigl{(}2\sqrt{s_{1}\ln(\lambda_{1}/\lambda)}\bigr{)},$
(53)

where $\theta(x)$ is the Heaviside step function. When $s_{0}=s_{1}$, this
reproduces the well-known result of the entanglement spectrum for 2D CFTs
Calabrese:2008cl .

As a comparison, the entanglement spectrum is expressed in terms an infinite
sum Bai:2022obp  (by the approach in Belin:2013dva )

$\rho(\lambda)=e^{s_{0}-s_{1}}\biggl{(}\delta(\lambda_{1}-\lambda)+\frac{\theta(\lambda_{1}-\lambda)}{\lambda}\sum_{i=0}^{\infty}{\frac{v_{i+1}}{i!}\bigl{(}\ln(\lambda_{1}/\lambda)\bigr{)}^{i}}\biggr{)},$
(54)

where the coefficients $\\{v_{i}\\}$ can be calculated order by order by
expanding

$\exp\biggl{(}\sum\limits_{i=1}^{\infty}{u_{i}n^{-i}}\biggr{)}=1+\sum\limits_{i=1}^{\infty}{v_{i}n^{-i}}.$
(55)

The sum (54) captures the low-lying part of the spectrum Bai:2022obp , where
“low-lying” represents the spectrum with lower energy, or with $\lambda$
closer to the largest eigenvalue Haldane:2008en .

For $\lambda\to 0$, the spectrum can be approximated by the saddle point
method. The saddle point $n_{0}$ is given by

$\frac{\partial}{\partial
n}\bigl{(}(n-1)S_{n}\bigr{)}\biggl{|}_{n_{0}}+\ln{\lambda}=0,$ (56)

and the integral (45) is approximated by

$\rho(\lambda\to
0)\sim\frac{1}{\lambda^{n+1}}e^{(1-n)S_{n}}\Bigl{[}2\pi\frac{\partial^{2}}{\partial
n^{2}}\bigl{(}(1-n)S_{n}\bigr{)}\Bigr{]}^{-1/2}\bigg{|}_{n_{0}}=\frac{1}{\lambda^{n+1}}e^{(1-n)S_{n}}\Bigl{[}2\pi\frac{C_{E}(n)}{n^{2}}\Bigr{]}^{-1/2}\bigg{|}_{n_{0}}.$
(57)

This shows a relation between the capacity of entanglement and the
entanglement spectrum at the vicinity of the saddle point.

##  4 Generating the potential and a byproduct

###  4.1 Generating $V(\phi)$ from the $V=0$ solution

Although the potential (5) the solution (10)–(13) are not new and were
rediscovered many times Gao:2004tu ; Gao:2004tv ; Gao:2005xv ; Anabalon:2012ta
; Feng:2013tza ; Faedo:2015jqa , we give a way to obtain them without any
manual input of unknown functions. We will show that the potential (5) can be
generated from the $V=0$ solution that was found in Horowitz:1991cd .

It was known that the potential can be generated from a given function in the
metric or the scalar field, and this method was used to construct many scalar
potentials Gubser:2008ny ; Li:2011hp ; Cai:2012xh ; Anabalon:2012ta ;
Feng:2013tza . Generally, all parameters in the metric will enter the
potential by this procedure. On the one hand, the parameters in the potential
are model parameters, i.e., parameters in the Lagrangian. On the other hand,
the solution parameters have integration constants such as mass and chemical
potential that are not expected in the potential Feng:2013tza . Therefore, it
is nontrivial for these solution parameters to not be in the potential. In
other words, requiring a parameter in the solution to not be in the potential
will give a constraint on the potential. An observation in Ren:2019lgw  is
that if we choose this parameter to be the spatial curvature $k$ with
reasonable assumptions, the potential will be completely determined. We
generalize the appendix A of Ren:2019lgw  to arbitrary dimensions in the
following.

We consider the neutral solution for simplicity. The action is

$S=\int
d^{d+1}x\,\sqrt{-g}\left(R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\right),$ (58)

where $V(\phi)$ is the potential of the scalar field $\phi$. We consider the
following metric ansatz:

$ds^{2}=e^{2\mathcal{A}(\bar{r})}(-h(\bar{r})dt^{2}+d\hskip
0.83328pt\Sigma_{d-1,k}^{2})+\frac{e^{2\mathcal{B}(\bar{r})}}{h(\bar{r})}d\bar{r}^{2}\,,$
(59)

where $\bar{r}$ is the AdS radial coordinate, and the metric for the
$(d-1)$-dimensional sphere, plane, and hyperbolic space can be written as

$d\Sigma_{d-1,k}^{2}=\frac{dx^{2}}{1-kx^{2}}+x^{2}d\Omega_{d-2}^{2}\,,$ (60)

where $d\Omega_{d-2}^{2}$ is a $(d-2)$-dimensional sphere of unit radius.
There is one gauge degree of freedom in the unknown functions
$\mathcal{A}(\bar{r})$, $\mathcal{B}(\bar{r})$, $h(\bar{r})$, and
$\phi(\bar{r})$, and it will be fixed by $\phi=\bar{r}$.

Equations of motion are obtained by the action (58) with the metric ansatz
(59). The Einstein’s equation gives

$\displaystyle\mathcal{A}^{\prime}\mathcal{B}^{\prime}=\frac{1}{2(d-2)}\phi^{\prime
2}+\mathcal{A}^{\prime\prime}\,,$ (61)
$\displaystyle(e^{d\mathcal{A}-\mathcal{B}}h^{\prime})^{\prime}+2(d-2)e^{(d-2)\mathcal{A}+\mathcal{B}}k=0\,.$
(62)

The first equation comes from $G_{tt}$ and $G_{\bar{r}\bar{r}}$, and the
second equation comes from $G_{\bar{r}\bar{r}}$ and $G_{xx}$. Solving the
potential from $G_{\bar{r}\bar{r}}$ gives

$V=\bigl{[}\frac{1}{2}\phi^{\prime 2}h-d(d-1)\mathcal{A}^{\prime
2}h-(d-1)\mathcal{A}^{\prime}h^{\prime}\bigr{]}e^{-2\mathcal{B}}+(d-1)(d-2)e^{-2\mathcal{A}}k\,.$
(63)

Other equations can be derived from (61), (62), and (63). Starting with a
given $\mathcal{A}(\bar{r})$, we can obtain $V(\bar{r})$ in the following way:

$\mathcal{A}\xrightarrow{\eqref{eq:B}}\mathcal{B}\xrightarrow{\eqref{eq:h}}h\xrightarrow{\eqref{eq:V}}V\,.$
(64)

The function $\mathcal{A}(\bar{r})$ plays the role of a generating function.
Finally, replacing the function $V(\bar{r})$ with $V(\phi)$ gives the
potential. A caveat is that only careful choices of $\mathcal{A}$ can we
obtain a relatively simple $V(\phi)$.

The potential $V$ solved by (64) generally depends on $k$. As in Ren:2019lgw ,
we require that the potential $V$ is independent of $k$, to have fewer
parameters than the solution. The terms dependent on $k$ must cancel:

$V=V^{(0)}+V^{(k)},\qquad V^{(k)}=0\,.$ (65)

We need an additional constraint: the function $h$ depends on $k$, and other
functions are independent of $k$. In other words, for a given potential $V$,
the only difference between the $k=0$ solution and the $k\neq 0$ solution is
some terms $h^{(k)}$ in $h$. The motivation for this constraint is that if it
is satisfied, the equation (62) for $h$ is a linear equation, which we can
take advantage of. The following will be based on the above constraints.

We decompose $h$ into a $k$-independent part and a $k$-dependent part:
$h(\bar{r})=h^{(0)}+h^{(k)}$, where $h^{(0)}$ is the solution of $h$ at $k=0$.
Similarly, we decompose the equation of motion into a part at $k=0$ and a part
dependent on $k$. The part at $k=0$ requires that $\mathcal{A}$,
$\mathcal{B}$, and $h^{(0)}$ satisfy the equations of motion with $V=V^{(0)}$,
and the part dependent on $k$ requires that $h^{(k)}$ satisfy the equations of
motion with $V=0$. As a consequence, the $V=V^{(0)}$, $k=0$ solution and the
$V=0$, $k=1$ solution share the same generating function $\mathcal{A}$ (as
well as $\mathcal{B}$ and $\phi$). Start with a solution with $V=0$ and $k=1$
as a seed, and we can use the procedure (64) with $h=1$ and $k=0$ to obtain
the potential $V(\phi)$.

The solution of $h$ from (62) is given by

$h=\int e^{-d\mathcal{A}+\mathcal{B}}\left(-2(d-2)k\int
e^{(d-2)\mathcal{A}+\mathcal{B}}d\bar{r}+C_{2}\right)d\bar{r}+C_{1}\,,$ (66)

where $C_{1}=C_{2}=0$ gives the solution for the system with $V=0$. The
general $V(\phi)$ solution will contain the two integration constants coming
from the second-order linear equation for $h$. If we take $C_{1}=1$ and
$C_{2}\propto k$, we can obtain the potential $V(\phi)$ as (5).

Now it boils down to solving the system with $V=0$. This is nontrivial, but
has been achieved in Horowitz:1991cd . (The AdS4 case was in Garfinkle:1990qj
earlier.) We shall not repeat the details. However, we use a more convenient
coordinate system to show that the equations of motion are solvable when
$V=0$. The metric ansätz is

$ds^{2}=-e^{2A}hdt^{2}+e^{2B}\biggl{(}\frac{dr^{2}}{h}+r^{2}d\Sigma_{d-1,k}^{2}\biggr{)}\,,$
(67)

with the gauge Duff:1996hp

$A+(d-2)B=0\,.$ (68)

The independent equations are

$\displaystyle\phi^{\prime 2}+2(d-1)\biggl{(}B^{\prime\prime}+(d-2)B^{\prime
2}+\frac{d-1}{r}B^{\prime}\biggr{)}=0\,,$ (69)
$\displaystyle\text{\small$h^{\prime\prime}-\biggl{(}2(d-1)-\frac{d-3}{r}\biggr{)}h^{\prime}-2\biggl{(}(d-1)B^{\prime\prime}+\frac{(d-1)^{2}}{r}B^{\prime}+\frac{(d-2)}{r^{2}}\biggr{)}h+\frac{2(d-2)k}{r^{2}}=0$}\,,$
(70) $\displaystyle
V=-e^{-2B}\biggl{(}h^{\prime\prime}+\frac{3d-5}{r}h^{\prime}+\frac{2(d-2)^{2}}{r^{2}}h-\frac{2(d-2)^{2}}{r^{2}}k\biggr{)}.$
(71)

The last equation shows that the function $h$ is easily solvable when $V=0$.
Then $B$ can be solved by the second equation, and $A$ by the first equation.

Once we obtained the metric, converting from the metric (67) to (59), we
obtain the generating function

$e^{\mathcal{A}}=b\left(e^{\frac{d-2}{(d-1)\alpha}\bar{r}}-e^{-\frac{\alpha}{2}\bar{r}}\right)^{-\frac{1}{d-2}}.$
(72)

Alternatively, we can also use the solution of $\phi(r)$ as the generating
function. Let $V_{\alpha}(\phi)$ be the potential (5) with parameter $\alpha$,
a more general potential than (5) is

$V(\phi)=V_{\alpha}+V_{\text{extra}},$ (73)

where $V_{\text{extra}}$ comes from a nonzero $C_{2}$ in (66) at $k=0$. This
potential has already been obtained in Feng:2013tza  by treating the solution
of $\phi(r)$ as an ansätz. Here, we treat the $V=0$ solution as a seed to
generate the general potential, and emphasize its naturalness. The extra terms
in (73) involve hypergeometric functions and look cumbersome. However, in the
AdS4 case, a six-exponential potential with a simple structure can be obtained
as (75) below.

###  4.2 Two neutral solutions with scalar hair in AdS4

It was observed in Ren:2019lgw  that there are two neutral limits of the
hyperbolic black holes (10)–(13): a trivial neutral limit $b=0$ where we
obtain the hyperbolic Schwarzschild-AdS black hole, and a nontrivial neutral
limit $c=0$ where we obtain a hairy black hole. As a consequence, the
hyperbolic black hole spontaneously develops a scalar hair below a critical
temperature. This was used to analytically study phase transitions of the
Rényi entropy Bai:2022obp . In the following, we obtain a different neutral
hyperbolic black hole with scalar hair for the same system (4) with (5).

Start with a more general action

$S=\int
d^{4}x\sqrt{-g}\biggl{(}R-\frac{1}{4}e^{-\alpha\phi}(F^{1})^{2}-\frac{1}{4}e^{\phi/\alpha}(F^{2})^{2}-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\biggr{)}.$
(74)

Let $V_{\alpha}(\phi)$ be the three-exponential potential (5), and we consider
the following six-exponential potential Ren:2019lgw 444Earlier works on
deriving a six-exponential potential were Anabalon:2012ta ; Feng:2013tza , and
various properties were studied in Anabalon:2019tcy ; Anabalon:2017yhv ;
Anabalon:2020pez , for example.

$V(\phi)=(1+\beta)V_{\alpha}(\phi)-\beta V_{-\alpha}(\phi)\,.$ (75)

The solution of the metric $g_{\mu\nu}$, gauge fields $A_{\mu}^{1,2}$, and
dilaton field $\phi$ is

$\displaystyle ds^{2}=-f(r)\hskip 0.83328ptdt^{2}+\frac{1}{f(r)}\hskip
0.83328ptdr^{2}+U(r)\hskip 0.83328ptd\Sigma_{2,k}^{2}\,,$ (76) $\displaystyle
A^{1}=2\gamma\sqrt{\frac{bc}{1+\alpha^{2}}}\left(\frac{1}{r_{h}}-\frac{1}{r}\right)dt\,,$
(77) $\displaystyle
A^{2}=2\alpha\sqrt{\frac{(\gamma^{2}-1)bc}{1+\alpha^{2}}}\left(\frac{1}{r_{h}-b}-\frac{1}{r-b}\right)dt\,,$
(78) $\displaystyle
e^{\alpha\phi}=\left(1-\frac{b}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}},$
(79)

with

$\displaystyle f$
$\displaystyle=\left(k-\frac{c}{r}\right)\left(1-\frac{b}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}+(1+\beta)\frac{r^{2}}{L^{2}}\biggl{(}1-\frac{b}{r}\biggr{)}^{\frac{2\alpha^{2}}{1+\alpha^{2}}}+(\gamma^{2}-1)\frac{bc}{r^{2}}\biggl{(}1-\frac{b}{r}\biggr{)}^{-\frac{2\alpha^{2}}{1+\alpha^{2}}}$
$\displaystyle\hskip
65.04034pt-\beta\frac{r^{2}}{L^{2}}\biggl{(}1+\frac{1-3\alpha^{2}}{1+\alpha^{2}}\,\frac{b}{r}+\frac{(1-\alpha^{2})(1-3\alpha^{2})}{(1+\alpha^{2})^{2}}\,\frac{b^{2}}{r^{2}}\biggr{)}\biggl{(}1-\frac{b}{r}\biggr{)}^{\frac{1-\alpha^{2}}{1+\alpha^{2}}},$
(80) $\displaystyle U$
$\displaystyle=r^{2}\biggl{(}1-\frac{b}{r}\biggr{)}^{\frac{2\alpha^{2}}{1+\alpha^{2}}}.$
(81)

The solution has parameters $b$, $c$ and $\gamma$ in addition to $\alpha$ and
$\beta$. By taking $\gamma=1$, we obtain an EMD system with a six-exponential
potential of the dilaton. The curvature singularity is at $r=0$ and $r=b$. The
mass is given by

$M=\frac{V_{\Sigma}}{8\pi
G}\biggl{(}c+k\frac{1-\alpha^{2}}{1+\alpha^{2}}b-\beta\frac{(1-\alpha^{2})(1-3\alpha^{2})(3-\alpha^{2})}{2(1+\alpha^{2})^{3}L^{2}}b^{3}\biggr{)}.$
(82)

The trivial neutral limit of this solution is $b=0$, and the scalar field
$\phi$ vanishes, and the potential becomes the cosmological constant. We are
interested in the nontrivial neutral limit $c=0$ later.

In Anabalon:2012ta ; Feng:2013tza  and Faedo:2015jqa , it was found that there
are two analytic solutions to the same system with potential
$V_{\alpha}(\phi)$, and this was explained in Nozawa:2022upa  (see also
Anabalon:2017yhv ). Here we give a simple explanation as follows. The six-
exponential potential has an additional parameter $\beta$. By taking
$\beta=0$, we obtain a solution for the three-exponential potential
$V_{\alpha}(\phi)$; by taking $\beta=-1$, $\alpha\to-\alpha$ and
$\phi\to-\phi$, we obtain a different solution for the same potential. Thus,
the nontrivial neutral limit $c=0$ gives two different hyperbolic black holes
with scalar hair. One neutral solution was studied in Ren:2019lgw ;
Bai:2022obp , and the other one is given by

$\displaystyle f(r)$
$\displaystyle=\biggl{[}-1+\frac{r^{2}}{L^{2}}\biggl{(}1+\frac{1-3\alpha^{2}}{1+\alpha^{2}}\,\frac{b}{r}+\frac{(1-\alpha^{2})(1-3\alpha^{2})}{(1+\alpha^{2})^{2}}\,\frac{b^{2}}{r^{2}}\biggr{)}\biggr{]}\biggl{(}1-\frac{b}{r}\biggr{)}^{\frac{1-\alpha^{2}}{1+\alpha^{2}}},$
(83) $\displaystyle\phi$
$\displaystyle=-\frac{2\alpha}{1+\alpha^{2}}\ln\left(1-\frac{b}{r}\right).$
(84)

The boundary condition for the scalar field corresponds to a multi-trace
deformation in the dual CFT Witten:2001ua . The boundary conditions
corresponding to the triple-trace deformation are different for the two
solutions, while boundary conditions corresponding to double-trace deformation
are the same for the two solutions.

##  5 Summary and discussion

In this paper, we have calculated the supersymmetric Rényi entropies (with a
spherical entangling surface) from a class of hyperbolic black holes with
scalar hair. The main conclusions are as follows:

  * •

By employing a class of hyperbolic black holes with scalar hair, we explicitly
obtain the holographic supersymmetric Rényi entropies. While the result
reproduces many known results, more results with more intriguing features are
also obtained.

  * •

We calculate the supersymmetric capacity of entanglement and showed that it
cannot be mapped to the heat capacity of hyperbolic black holes due to the
fact that the BPS condition gives a constraint between the temperature and the
chemical potential.

  * •

From the Rényi entropies that are analytic at $n=\infty$, we calculate the
entanglement spectrum as convolutions of generalized hypergeometric functions.

  * •

We show that the potential of the EMD system can be generated from the
$V(\phi)=0$ solution.

  * •

There are two nontrivial neutral limits of the EMD system, giving hyperbolic
black holes with scalar hair. Scalar condensation may happen at sufficiently
low temperatures.

The following topics need further investigation: (i) The CFT calculations for
the models in table 1. (ii) Violation of inequalities when
$\alpha>\alpha_{*}$, especially the 1-charge black hole in AdS4 as a top-down
special case. (iii) The geometric interpretation of the supersymmetric Rényi
entropy. (iv) Whether there are phase transitions for this class of hyperbolic
black hole solutions.

###### Acknowledgements.

J.R. thanks Xiaoxuan Bai, Antonio Gallerati, Song He, Masato Nozawa, Yi Pang,
Chiara Toldo, and Yang Zhou for helpful communications. This work was
supported in part by the NSF of China under Grant No. 11905298, and
Fundamental Research Funds for the Central Universities, Sun Yat-sen
University under Grant No. 23qnpy61.

##  Appendix A Special cases of $D=4,5,6,7$ supergravities

In STU supergravities, there are U$(1)^{4}$ gauge fields in AdS4, U$(1)^{3}$
gauge fields in AdS5, and U$(1)^{2}$ gauge fields in AdS7 Cvetic:1999xp .
Special cases of them can be reduced to EMD systems. They are 1-charge,
2-charge, and 3-charge black holes in AdS4; 1-charge and 2-charge black holes
in AdS5; and 1-charge black hole in AdS7.

The AdS4 Lagrangian is

$\mathcal{L}=R-\frac{1}{2}(\partial\vec{\phi})^{2}+8g^{2}(\cosh\phi_{1}+\cosh\phi_{2}+\cosh\phi_{3})-\frac{1}{4}\sum_{i=1}^{4}e^{\vec{a}_{i}\cdot\vec{\phi}}(F^{i}_{(2)})^{2}\,,$
(85)

where $\vec{\phi}=(\phi_{1},\phi_{2},\phi_{3})$, $\vec{a}_{1}=(1,1,1)$,
$\vec{a}_{2}=(1,-1,-1)$, $\vec{a}_{3}=(-1,1,-1)$, and $\vec{a}_{4}=(-1,-1,1)$.
More details can be found in Cvetic:1999xp . The solution is given by
Duff:1999gh ; Sabra:1999ux

$\displaystyle ds^{2}$
$\displaystyle=-(H_{1}H_{2}H_{3}H_{4})^{-1/2}fdt^{2}+(H_{1}H_{2}H_{3}H_{4})^{1/2}(f^{-1}dr^{2}+r^{2}d\hskip
0.83328pt\Sigma_{2,k}^{2})\,,$ (86) $\displaystyle X_{i}$
$\displaystyle=H_{i}^{-1}(H_{1}H_{2}H_{3}H_{4})^{1/4}\,,$ (87) $\displaystyle
A_{(1)}^{i}$ $\displaystyle=\sqrt{k}(1-H_{i}^{-1})\coth\beta_{i}\hskip
0.83328ptdt\,,$ (88)

with $X_{i}=e^{-\frac{1}{2}\vec{a}_{i}\cdot\vec{\phi}}$, and

$f=k-\frac{\mu}{r}+4g^{2}r^{2}(H_{1}H_{2}H_{3}H_{4})\,,\qquad
H_{i}=1+\frac{\mu\sinh^{2}\beta_{i}}{kr}\,.$ (89)

We call this general solution (1+1+1+1)-charge black hole, or 4-charge black
hole in AdS4 if there is no confusion. For special cases, the following naming
convention is used in the literature.

$H_{i}$ ($i=1,2,3,4$) | Name
---|---
$H_{1}=H_{2}=H_{3}$, $H_{4}\quad(\neq 1)$ | (3+1)-charge black hole in AdS4
$H_{1}=H_{2}$, $H_{3}=H_{4}\quad(\neq 1)$ | (2+2)-charge black hole in AdS4
$H_{1}=H_{2}=H_{3}=H$, $H_{4}=1$ | 3-charge black hole in AdS4
$H_{1}=H_{2}=H$, $H_{3}=H_{4}=1$ | 2-charge black hole in AdS4
$H_{1}=H$, $H_{2}=H_{3}=H_{4}=1$ | 1-charge black hole in AdS4
$H_{1}=H_{2}=H_{3}=H_{4}=H$ | RN-AdS4 black hole

The AdS5 Lagrangian is

$\mathcal{L}=R-\frac{1}{2}(\partial\vec{\varphi})^{2}+4g^{2}\sum_{i}X_{i}^{-1}-\frac{1}{4}\sum_{i=1}^{4}X_{i}^{-2}(F^{i}_{(2)})^{2}\,.$
(90)

The solution is Behrndt:1998jd

$\displaystyle
ds^{2}=-(H_{1}H_{2}H_{3})^{-2/3}fdt^{2}+(H_{1}H_{2}H_{3})^{1/3}(f^{-1}dr^{2}+r^{2}d\hskip
0.83328pt\Sigma_{3,k}^{2})\,,$ (91) $\displaystyle
X_{i}=H_{i}^{-1}(H_{1}H_{2}H_{3})^{1/3}\,,$ (92) $\displaystyle
A_{(1)}^{i}=\sqrt{k}(1-H_{i}^{-1})\coth\beta_{i}dt\,,$ (93)

with

$f=k-\frac{\mu}{r^{2}}+g^{2}r^{2}(H_{1}H_{2}H_{3})\,,\qquad
H_{i}=1+\frac{\mu\sinh^{2}\beta_{i}}{kr^{2}}\,.$ (94)

We call this general solution (1+1+1)-charge black hole, or 3-charge black
hole in AdS5 if there is no confusion. For special cases, we have (2+1)-charge
black hole, 2-charge black hole, 1-charge black hole in AdS5. When
$H_{1}=H_{2}=H_{3}$, we have the RN-AdS5 black hole.

The static AdS6 black hole metric is Chow:2011fh ; Hosseini:2019and

$\displaystyle
ds^{2}=-\frac{9}{2}(H_{1}H_{2})^{-3/4}fdt^{2}+(H_{1}H_{2})^{1/4}(f^{-1}dr^{2}+r^{2}d\hskip
0.83328pt\Sigma_{4,k}^{2}),,$ (95) $\displaystyle
X_{i}=H_{i}^{-1}(H_{1}H_{2})^{3/8}\,,$ (96) $\displaystyle
A_{(1)}^{i}=\sqrt{k}\coth\beta_{i}(1-H_{i}^{-1})dt\,,$ (97)

with

$f(r)=k-\frac{\mu}{r^{3}}+\frac{2}{9}r^{2}H_{1}H_{2}\,,\qquad
H_{i}=1+\frac{\mu\sinh^{2}\beta_{i}}{kr^{3}}\,.$ (98)

Here, $i=1,2$. We call this solution (1+1)-charge black hole in AdS6. Special
cases are 2-charge black hole and 1-charge black hole in AdS6. Note that the
2-charge case ($H_{1}=H_{2}$) is not the RN-AdS6 black hole.

The AdS7 Lagrangian is

$e^{-1}\mathcal{L}=R-\frac{1}{2}(\partial\vec{\varphi})^{2}-g^{2}V-\frac{1}{4}\sum_{i=1}^{2}e^{\vec{a_{i}}\cdot\vec{\varphi}}(F^{i}_{(2)})^{2}\,.$
(99)

The solution is Cvetic:1999xp

$\displaystyle
ds_{7}^{2}=-(H_{1}H_{2})^{-4/5}fdt^{2}+(H_{1}H_{2})^{1/5}(f^{-1}dr^{2}+r^{2}d\Omega_{5,k}^{2})\,,$
(100) $\displaystyle X_{i}=H_{i}^{-1}(H_{1}H_{2})^{2/5}\,,$ (101)
$\displaystyle A_{(1)}^{i}=\sqrt{k}\coth\beta_{i}(1-H_{i}^{-1})dt\,,$ (102)

with

$f=k-\frac{\mu}{r^{4}}+\frac{1}{4}g^{2}r^{2}H_{1}H_{2}\,,\qquad
H_{i}=1+\frac{\mu\sinh^{2}\beta_{i}}{kr^{4}}\ ,$ (103)

We call this solution (1+1)-charge black hole in AdS7. Special cases are
2-charge black hole and 1-charge black hole in AdS7. Note that the 2-charge
case ($H_{1}=H_{2}$) is not the RN-AdS7 black hole.

For EMD truncations of these solutions, we need to shift the radial coordinate
$r\to r-\mu\sinh^{2}\beta$ to obtain the solutions (10)–(13).

##  Appendix B Fayet-Iliopoulos gauged supergravity

We briefly review the $\mathcal{N}=2$, $D=4$ supergravity with Abelian Fayet-
Iliopoulos (FI) gaugings that can be reduced to EMD systems. See
Nozawa:2022upa  for more details and references. For $n_{V}$ number of abelian
vector multiplets Andrianopoli:1996cm , the model describes $n_{V}+1$ vector
fields $A_{\mu}^{I}$($I=0,1,\dots,n_{V}$) and $n_{s}=n_{V}$ complex scalars
fields $z^{\alpha}(\alpha=1,\dots,n_{s})$. These scalars parametrize an
$n_{V}$-dimensional Hodge-Kähler manifold, which is the base of a symplectic
bundle with covariantly holomorphic section

$\mathcal{V}=\biggl{(}\begin{array}[]{c}X^{I}\\\
F_{I}\end{array}\biggr{)}\,,\qquad\mathcal{D}_{\bar{\alpha}}\mathcal{V}=\partial_{\bar{\alpha}}\mathcal{V}-\frac{1}{2}(\partial_{\bar{\alpha}}\mathcal{K})\mathcal{V}=0\,,$
(104)

where $\mathcal{V}$ obeys the symplectic constraint
$\langle\mathcal{V},\bar{\mathcal{V}}\rangle\equiv
X^{I}\bar{F}_{I}-F_{I}\bar{X}^{I}=i$ and
$\langle\mathcal{V},\partial_{\alpha}{\mathcal{V}}\rangle=0$;
$\mathcal{K}=\mathcal{K}(z^{\alpha},\bar{z}^{\alpha})$ is the Kähler
potential, and $\mathcal{D}_{\alpha}$ denotes the Kähler covariant derivative.
Writing

$\mathcal{V}=e^{K/2}v,\qquad v=\biggl{(}\begin{array}[]{c}Z^{I}\\\
\frac{\partial}{\partial Z^{I}}F(Z)\end{array}\biggr{)},$ (105)

where $v$ is the holomorphic symplectic vector. In appropriate symplectic
frame, we assume the existence of prepotential $F$ that is a homogeneous
function of degree two.

The bosonic gauged Lagrangian is

$\mathcal{L}=\frac{1}{2}(R-2V)\star
1-g_{\alpha\bar{\beta}}dz^{\alpha}\wedge\star
d\bar{z}^{\bar{\beta}}+\frac{1}{2}I_{IJ}F^{I}\wedge\star
F^{J}+\frac{1}{2}R_{IJ}F^{I}\wedge F^{J}\,.$ (106)

where the $n_{V}+1$ vector field strengths are $F^{I}=dA^{I}$; $I_{IJ}={\rm
Im}\mathcal{N}_{IJ}$, $R_{IJ}={\rm Re}\mathcal{N}_{IJ}$, where
$\mathcal{N}_{IJ}$ is defined by the relations $F_{I}=\mathcal{N}_{IJ}X^{J}$
and
$\mathcal{D}_{\bar{\alpha}}\bar{F}_{I}=\mathcal{N}_{IJ}\mathcal{D}_{\bar{\alpha}}\bar{X}^{J}$.
The scalar potential is

$V=-2g_{I}g_{J}\left(I^{IJ}+8\bar{X}^{I}X^{J}\right)\,,$ (107)

where $I^{IJ}$ is the inverse of $I_{IJ}$, and $g_{I}$ is the FI coupling
constants.

We consider the following prepotential of $\mathcal{N}=2$ supergravity with
one complex scalar ($n_{V}=1$):

$F(X)=-\frac{i}{4}(X^{0})^{n}(X^{1})^{2-n}\,.$ (108)

The values of the parameter $n=1,1/2$, and $3/2$ correspond to special cases
of STU supergravity. This is a truncation of the STU model with the
prepotential

$F_{\rm STU}(X)=-\frac{i}{4}\sqrt{X^{0}X^{1}X^{2}X^{3}}\,.$ (109)

Setting $Z^{0}=1$ and $Z^{1}=z$, the symplectic vector is
$v=(1,z,-\frac{i}{4}nz^{2-n},-\frac{i}{4}(2-n)z^{1-n})^{T}$. The system (74)
is obtained by further truncating the theory to a single real scalar
$z=\bar{z}$ and the purely electrically charged case $F^{I}\wedge F^{J}=0$.

##  Appendix C The IR geometry for $\alpha=\alpha_{*}$

In the special case $\alpha=\alpha_{*}:=(d-2)\sqrt{\frac{2}{d-1}}$, the black
hole under the BPS condition has a distinctive IR geometry at zero
temperature: it has curvature singularity. As a comparison, when
$0\leq\alpha<\alpha_{*}$, the IR geometry at zero temperature is
AdS${}_{2}\times\mathbb{H}^{d-1}$, i.e., it has a degenerate horizon.

When $\alpha=\alpha_{*}$, the IR limit of the geometry at zero temperature is

$ds^{2}=(r-1)^{\frac{2}{d-1}}\biggl{(}-(r-1)dt^{2}+\frac{dr^{3}}{(r-1)^{3}}+d\Sigma_{d-1}^{2}\biggr{)}.$
(110)

By the change of variables

$\tilde{r}=(r-1)^{-1/2},$ (111)

the IR geometry is written as

$ds^{2}=\tilde{r}^{\frac{2\theta}{d-1}}\biggl{(}-\frac{dt^{2}}{\tilde{r}^{2\mathsf{z}}}+\frac{d\tilde{r}^{2}+d\Sigma_{d-1}^{2}}{\tilde{r}^{2}}\biggr{)},$
(112)

which is a hyperscaling-violating geometry with the spatial part being
$\mathbb{H}^{d-1}$. The Lifshitz scaling exponent $\mathsf{z}$ and the
hyperscaling violation exponent $\theta$ are

$\mathsf{z}=2,\qquad\theta=d-3.$ (113)

In particular, when $d=3$, it is a Lifshitz geometry with the spatial part
being $\mathbb{H}^{d-1}$.

It is interesting to make a comparison to the planar black hole solutions of
the same EMD system (4). The IR geometries of extremal planar black holes are
Ren:2019lgw :

  * •

$0<\alpha<(d-2)\sqrt{\frac{2}{d(d-1)}}$. The IR geometry is
AdS${}_{2}\times\mathbb{R}^{d-1}$.

  * •

$\alpha=(d-2)\sqrt{\frac{2}{d(d-1)}}$. The IR geometry is conformal to
AdS${}_{2}\times\mathbb{R}^{d-1}$ Gubser:2012yb ; Gouteraux:2014hca .

  * •

$\alpha>(d-2)\sqrt{\frac{2}{d(d-1)}}$. The extremal limit of the EMD system
(4) is the same as an Einstein-scalar system. The IR geometry is a
hyperscaling-violating geometry.

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