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

# Holographic Rényi entropies from hyperbolic black holes with scalar hair

Xiaoxuan Bai and Jie Ren<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Rényi entropies as a generalization of the entanglement entropy imply much
more information. We analytically calculate the Rényi entropies (with a
spherical entangling surface) by means of a class of neutral hyperbolic black
holes with scalar hair as a one-parameter generalization of the MTZ black
hole. The zeroth-order and third-order phase transitions of black holes lead
to discontinuity of the first and second derivatives of the Rényi entropies,
respectively. From the Rényi entropies which are analytic at $n=\infty$, we
can express the entanglement spectrum as an infinite sum in terms of the Bell
polynomial. We show that the analytic treatment is in agreement with numerical
calculations for the low-lying entanglement spectrum in a wide range of
parameters.

##  1 Introduction

Entanglement entropy, which measures the entanglement of different parts in a
quantum system, has considerable applications in condensed matter physics,
e.g., the detection of topological ordered phases Levin:2006lw ; Kitaev:2005dm
, quantum phase transitions Vidal:2002rm  and fractional quantum Hall effects
Haldane:2008en  (see Amico:2007ag ; Laflorencie:2015eck  for reviews).
Moreover, the holographic description of the entanglement entropy Ryu:2006bv ;
Ryu:2006ef  in terms of the AdS/CFT correspondence may inspire a deeper
understanding of the quantum gravity VanRaamsdonk:2010pw . As a generalization
of the entanglement entropy, the Rényi entropies Renyi:1961me  imply much more
information, e.g., with the Rényi entropies of all orders, the entanglement
spectrum Haldane:2008en  (eigenvalue distribution of the reduced density
matrix) can be determined Calabrese:2008cl ; Calabrese:2009qy ; Hung:2011nu ;
Belin:2013dva ; Guo:2020roc .

A quantum system can be described by a density operator $\rho$. For
simplicity, we consider a pure state consisting of two subsystems $A$ and $B$,
characterized by vectors in Hilbert spaces $H_{A}$ and $H_{B}$, respectively.
It is convenient to define the reduced density operator
$\rho_{A}=\text{Tr}_{B}\,\rho_{AB}$, which only depends on the degrees of
freedom of $A$. The entanglement (von Neumann) entropy is
$S_{EE}=-\text{Tr}\rho_{A}\log\rho_{A}$, which becomes zero when $\rho_{A}$
describes a pure state, or equivalently, the whole system can be characterized
by a product state. With $n$ as a parameter, the Rényi entropies are defined
as

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

which return to the entanglement entropy in the $n\rightarrow 1$ limit.

Generally, these entropies can be calculated with the replica trick
Holzhey:1994we ; Calabrese:2004eu ; Calabrese:2009qy  in quantum field theory
by considering the Euclidean functional integral on $n$ copies of Riemann
surfaces which are matched together. In terms of the AdS/CFT correspondence,
the entanglement entropy can be obtained from general relativity by the Ryu-
Takayanagi (RT) formula Ryu:2006bv ; Ryu:2006ef . In particular, for two
regions separated by a spherical entangling surface, the Rényi entropies of
the ground state can be calculated by means of a hyperbolic (topological)
black hole Casini:2011kv ; Hung:2011nu  (we briefly review this approach in
section 1.1). The temperature of the black hole is related to the Rényi
parameter $n$, where a larger $n$ corresponds to a lower temperature.

The hyperbolic Schwarzschild-AdS (SAdS) solution is

$ds^{2}=-\biggl{(}-1-\frac{2M}{r^{d-2}}+\frac{r^{2}}{L^{2}}\biggr{)}dt^{2}+\biggl{(}-1-\frac{2M}{r^{d-2}}+\frac{r^{2}}{L^{2}}\biggr{)}^{-1}dr^{2}+r^{2}d\Sigma_{d-1}^{2},$
(2)

where $d\Sigma_{d-1}^{2}=du^{2}+\sinh^{2}{u}\,d\Omega_{d-2}^{2}$ is a
$(d-1)$-dimensional hyperbolic space of unit radius. The hyperbolic SAdS black
hole is stable at any temperature Birmingham:1998nr . However, if a scalar
field is introduced to the system, the black hole may experience a phase
transition to the one with scalar hair at a critical temperature. This can be
implied from the fact that the IR geometry of the extremal hyperbolic black
hole (2) includes an AdS2 factor, which leads to the Breitenlohner-Freedman
bound violated Dias:2010ma . Accordingly, the Rényi entropies also have a
phase transition at a special $n$, which has been discussed in Belin:2013dva ;
Fang:2016ehk  in terms of numerical solutions of hairy hyperbolic black holes.
In this paper, we provide more analytic calculations.

The Einstein-Maxwell-dilaton (EMD) system, with gauge field and dilaton field
coupled to gravity, has wide applications in gauge/gravity duality. We start
with a class of analytic solutions of the EMD system in Gao:2004tu , whose
special cases correspond to gauge supergravities. In a particular neutral
limit, the gauge field vanishes while the dilaton field is kept Ren:2019lgw ,
resulting in neutral hyperbolic black holes with scalar hair. These black
holes have zeroth-order and third-order phase transitions, and the first and
second derivatives of the Rényi entropies show discontinuity, respectively.

With the analytic solutions in hand, we explicitly obtain the Rényi entropies
$S_{n}$ and calculate the entanglement spectrum. These Rényi entropies
converge to the same entanglement entropy in the $n\rightarrow 1$ limit. For
special cases, we find the Rényi entropies have the same $n$-dependence as the
universal result in 2D CFT. Starting with the analytic Rényi entropies, we can
calculate the entanglement spectrum by an inverse Laplace transform.
Consequently, we can express the entanglement spectrum in terms of an infinite
sum. By truncating this sum, we can verify that it is consistent with the
numerical calculation for a wide range of eigenvalues.

This paper is organized as follows. In section 2, we begin by introducing a
class of neutral hyperbolic black holes with scalar hair Ren:2019lgw  in AdS4
spacetime and refining their thermodynamics and stability analysis. Then we
calculate holographic Rényi entropies and discuss corresponding phase
transitions. Besides, we verify some Rényi entropy inequalities in terms of
the thermodynamically preferred solution. In section 3, we generalize these
calculations to higher dimensions. In section 4, we calculate the entanglement
spectrum from the Rényi entropies as an infinite sum in terms of the Bell
polynomial and discuss its validity by comparing it with additional numerical
calculations. In appendix A, we give the boundary counter terms for
holographic renormalization, from which we obtain the mass and free energy of
the hyperbolic black holes. In appendix B, we give a brief note on the
mathematics related to our calculations. We first review the inverse Laplace
transform and some numerical methods. Then we introduce the exponential Bell
polynomial.

###  1.1 Rényi entropies from hyperbolic black holes

For the case with a spherical entangling surface, we can calculate the Rényi
entropies by conformally mapping the causal development of the enclosed region
to a hyperbolic cylinder $\mathbb{R}\times\mathbb{H}^{d-1}$ (see Casini:2011kv
; Hung:2011nu  for details). Assuming that the radius of the entangling sphere
is $L$, then the corresponding hyperbolic space $\mathbb{H}^{d-1}$ also has
the radius $L$. Under this conformal transformation, the reduced density
matrix is acted by a unitary operator

$\rho=U\frac{e^{-H/T_{0}}}{Z(T_{0})}U^{-1},$ (3)

where $H$ is the modular Hamiltonian generating the (Rindler) time translation
on $\mathbb{R}\times\mathbb{H}^{d-1}$, and $Z(T_{0})=\text{Tr}[e^{H/T_{0}}]$
is the thermal partition function at temperature $T_{0}=1/2\pi L$. Thus,

$\text{Tr}[\rho^{n}]=\frac{Z(T_{0}/n)}{Z(T_{0})^{n}},$ (4)

from which the Rényi entropies can be calculated from the thermodynamics of
the CFT living on the hyperbolic cylinder $\mathbb{R}\times\mathbb{H}^{d-1}$.
In terms of the AdS/CFT correspondence, the partition function $Z(T_{0}/n)$ is
identified with that of a hyperbolic black hole at temperature $T=T_{0}/n$.
Thus, the Rényi entropies can be calculated by means of the hyperbolic black
hole. Furthermore, with the free energy $F=-T\log Z$ of black holes, we can
rewrite the Rényi entropies as

$S_{n}=\frac{n}{1-n}\frac{1}{T_{0}}\bigl{[}F(T_{0})-F(T_{0}/n)\bigr{]}.$ (5)

Equivalenty, with the relation $S_{\text{therm}}=-\partial F/\partial T$, we
can obtain

$S_{n}=\frac{n}{n-1}\frac{1}{T_{0}}\int_{T_{0}/n}^{T_{0}}{S_{\text{therm}}(T)dT}.$
(6)

##  2 Holographic Rényi entropies from hyperbolic AdS4 black holes

###  2.1 Neutral hyperbolic black holes with scalar hair in AdS4 spacetime

To study holographic Rényi entropies, we employ a class of neutral hyperbolic
black holes with scalar hair Ren:2019lgw  as a one-parameter family
generalization of the MTZ black hole Martinez:2004nb . The action is

$S=\int
d^{4}x\sqrt{-g}\Bigl{[}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\Bigr{]},$ (7)

with the potential (first found in Gao:2004tu )

$V(\phi)=-\frac{2}{\left(1+\alpha^{2}\right)L^{2}}\Bigl{[}\alpha^{2}\left(3\alpha^{2}-1\right)e^{-\phi/\alpha}+8\alpha^{2}e^{\left(\alpha-1/\alpha\right)\phi/2}+\left(3-\alpha^{2}\right)e^{\alpha\phi}\Bigr{]},$
(8)

where $\alpha$ is a parameter related to the potential, which vanishes when
$\alpha=0$ or $\alpha\rightarrow\infty$. The cases when $\alpha=0,\
1/\sqrt{3},\ 1$, and $\sqrt{3}$ correspond to special cases of supergravities.
Since the $\phi\rightarrow 0$ behavior is
$V(\phi)=-6/L^{2}-(1/L^{2})\phi^{2}+\mathcal{O}(\phi^{4})$, the mass of the
scalar field satisfies $m^{2}L^{2}=-2$. From the relation
$m^{2}L^{2}=\Delta(\Delta-d)$, the scaling dimension of the dual operator in
the CFT is $\Delta_{+}=2$ or $\Delta_{-}=1$.111The asymptotic behavior of the
scalar field near the AdS boundary is determined by its mass. If we use
Fefferman-Graham (FG) coordinates, the asymptotic expansion is
$\phi(x,z)=z^{\Delta_{-}}(\phi_{a}+\phi_{b}z+\cdots)$ for the alternative
quantization. We consider a sourceless boundary condition corresponding to a
triple-trace deformation in the dual CFT Witten:2001ua . Specifically, the
boundary condition for the following solution is given by
$\phi_{b}/\phi_{a}^{2}=\tau$, where $\tau=-(1-\alpha^{2})/4\alpha$.

The solution to the system is Ren:2019lgw

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+U(r)d\Sigma_{2}^{2}\ ,\qquad
e^{\alpha\phi}=\biggl{(}1-\frac{b}{r}\biggr{)}^{\frac{2\alpha^{2}}{1+\alpha^{2}}},$
(9)

with

$f(r)=-\biggl{(}1-\frac{b}{r}\biggr{)}^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}+\frac{r^{2}}{L^{2}}\left(1-\frac{b}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}},\qquad
U(r)=r^{2}\left(1-\frac{b}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}.$ (10)

This solution was obtained by taking a nontrivial neutral limit222There is
also a trivial neutral limit which gives the SAdS black hole Ren:2019lgw .
from a class of charged dilaton black hole solutions in Gao:2004tu . The
horizon radius of black holes is obtained by taking $f(r_{h})=0$, and the
parameter $b$ can be replaced by $r_{h}$ with

$b=r_{h}\Bigl{[}1-\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{2(1+\alpha^{2})}{1-3\alpha^{2}}}\Bigr{]}.$
(11)

Now we refine the analysis of the black hole thermodynamics studied in
Ren:2019lgw . The temperature is

$T=\frac{f^{\prime}(r_{h})}{4\pi}=\frac{1}{4\pi(1+\alpha^{2})L}\biggl{[}(3-\alpha^{2})\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{1+\alpha^{2}}{1-3\alpha^{2}}}-(1-3\alpha^{2})\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{-\frac{1+\alpha^{2}}{1-3\alpha^{2}}}\biggr{]},$
(12)

from which we can solve $r_{h}$ in terms of temperature as

$r_{h,\pm}=\Biggl{(}\frac{2\pi
L(1+\alpha^{2})T\pm\sqrt{(\alpha^{2}-3)(3\alpha^{2}-1)+4\pi^{2}L^{2}(1+\alpha^{2})^{2}T^{2}}}{\alpha^{2}-3}\Biggr{)}^{\frac{1-3\alpha^{2}}{1+\alpha^{2}}}L,$
(13)

which shows that there may be two solutions of black holes at a given
temperature, and the horizon radius is distinguished by $r_{h,\pm}$. More
specifically, when $0\leq\alpha\leq 1/\sqrt{3}$, only $r_{h,+}$ is real; when
$\alpha\geq\sqrt{3}$, only $r_{h,-}$ is real; when
$1/\sqrt{3}<\alpha<\sqrt{3}$, both $r_{h,\pm}$ are real. The two solutions can
be seen in figure 1 below. The temperature can reach zero when $\alpha\leq
1/\sqrt{3}$ or $\alpha\geq\sqrt{3}$. However, when
$1/\sqrt{3}<\alpha<\sqrt{3}$, there is a minimum temperature above zero:

$T_{\text{m}}=\frac{\sqrt{(3-\alpha^{2})(3\alpha^{2}-1)}}{2\pi(1+\alpha^{2})L}\leq
T_{0},$ (14)

at which the two solutions intersect. In a special case $\alpha=1$, the
$T_{\text{m}}$ has a maximum value $T_{0}=1/2\pi L$.

Figure 1: The free energy as a function of temperature for different values of
$\alpha$. The dashed line is for the SAdS black hole, while the solid line is
for the hairy black hole, where the blue one is for $r_{h,+}$ and the orange
one is for $r_{h,-}$. The subfigure describes the relation between the
temperature and horizon radius. In all figures, we have set
$L=G=V_{\Sigma}=1$.

The Bekenstein-Hawking entropy is given by

$S=\frac{V_{\Sigma}}{4G}U(r_{h})=\frac{V_{\Sigma}}{4G}\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{2(1-\alpha^{2})}{1-3\alpha^{2}}}L^{2},$
(15)

where $V_{\Sigma}$ is the (regulated) area of the hyperbolic space
$d\Sigma_{2}^{2}$. With a boundary condition corresponding to a triple-trace
deformation in the dual CFT Ren:2019lgw ; Caldarelli:2016nni , the mass
derived from the holographic renormalization is

$M=-\frac{V_{\Sigma}}{8\pi
G}\frac{1-\alpha^{2}}{1+\alpha^{2}}b=-\frac{V_{\Sigma}}{8\pi
G}\frac{1-\alpha^{2}}{1+\alpha^{2}}r_{h}\Bigl{[}1-\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{2(1+\alpha^{2})}{1-3\alpha^{2}}}\Bigr{]}.$
(16)

All thermodynamic quantities above can be expressed as functions of
temperature by means of (13). So we can easily verify the first law of
thermodynamics in the canonical ensemble: $dM=TdS$. Finally, the Helmholtz
free energy is

$F=M-TS=-\frac{V_{\Sigma}}{16\pi
G}r_{h}\biggl{[}1+\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{2(1+\alpha^{2})}{1-3\alpha^{2}}}\biggr{]}.$
(17)

A more rigorous treatment of the black hole thermodynamics is in appendix A,
in which the boundary counter terms are given together with the boundary
condition of the scalar field.

The stability of black holes can be analyzed by comparing the free energy of
black holes with and without scalar hair. Note that when $\alpha=0$, the black
hole is the SAdS black hole. For representative cases of $\alpha$, we plot the
free energy as a function of temperature in figure 1. The SAdS black hole is
always more stable than the hairy black hole at $T>T_{0}$. As the temperature
is decreased, we draw the following conclusions on phase transitions:

  * •

$0<\alpha<1$: the SAdS black hole experiences a third-order phase transition
to the hairy one with the horizon size $r_{h,+}$ at $T=T_{0}$.

    * •

$0<\alpha\leq 1/\sqrt{3}$: There is only one hairy black hole corresponding to
$r_{h,+}$.

    * •

$1/\sqrt{3}<\alpha<1$: Besides, there is another unstable branch of hairy
black holes ($r_{h,-}$). A zeroth-order phase transition to the SAdS black
hole occurs at $T_{\text{m}}$.

  * •

$\alpha=1$: the SAdS black hole is always more stable than the hairy black
hole.

  * •

$\alpha>1$: the SAdS black hole experiences a third-order phase transition to
the hairy one with the horizon size $r_{h,-}$ at $T=T_{0}$.

    * •

$1<\alpha<\sqrt{3}$: The stable hairy black hole corresponds to $r_{h,-}$,
while an unstable branch corresponds to $r_{h,+}$. A zeroth-order phase
transition to the SAdS black hole occurs at $T_{\text{m}}$.

    * •

$\alpha\geq\sqrt{3}$: There is only one hairy black hole corresponding to
$r_{h,-}$

###  2.2 Holographic Rényi entropies

The expressions (5) and (6) describe the Rényi entropies without phase
transitions. So they have to be modified when we consider phase transitions,
e.g., for the zeroth-order phase transition in our cases, the Rényi entropies
are

$S_{n}(n\geq
1)=\frac{n}{1-n}\frac{1}{T_{0}}\bigl{(}F_{\text{H}}(T_{0})-F_{\text{H}}(T_{\text{m}})+F_{\text{S}}(T_{\text{m}})-F_{\text{S}}(T_{0}/n)\bigr{)},$
(18)

where $F_{\text{H}}$ is the free energy of the black hole with scalar hair,
$F_{\text{S}}$ is the free energy of the SAdS black hole, and $T_{\text{m}}$
is the phase transition temperature in (14). Note that (18) describes the
Rényi entropies with $n\geq 1$, because the most stable black hole is always
the SAdS one when $n<1$ (see figure 1). The Rényi entropies with $n<1$ are
described by (6), where $S_{\text{therm}}$ is the thermal entropy of the SAdS
black hole.

We calculate the Rényi entropies from the free energy (5) (the thermal entropy
calculation (6) gives the same result). For the cases when $\alpha\leq
1/\sqrt{3}$ and $\alpha\geq\sqrt{3}$, the phase transition only occurs at
$n=1$, so the Rényi entropies ($n\geq 1$) are

$S_{n}=\frac{nL^{2}}{8(n-1)G}\Bigl{[}2-\mathcal{S}_{\pm}(\alpha,n)^{\frac{1-3\alpha^{2}}{1+\alpha^{2}}}-\mathcal{S}_{\pm}(\alpha,n)^{\frac{3-\alpha^{2}}{1+\alpha^{2}}}\Bigr{]}V_{\Sigma}\,,$
(19)

with

$\mathcal{S}_{\pm}(\alpha,n)=\frac{1}{(3-\alpha^{2})n}\Bigl{[}(\alpha^{2}+1)\pm\sqrt{(\alpha^{2}+1)^{2}+(3-\alpha^{2})(1-3\alpha^{2})n^{2}}\Bigr{]},$
(20)

where $\mathcal{S}_{\pm}$ corresponds to the two solutions $r_{h,\pm}$:
$\mathcal{S}_{+}$ is for $0\leq\alpha\leq 1/\sqrt{3}$ and $\mathcal{S}_{-}$ is
for $\alpha\geq\sqrt{3}$. When $1/\sqrt{3}<\alpha<\sqrt{3}$, the Rényi
entropies are given by

$\bar{S}_{n}=\begin{cases}S_{n}(\alpha=0)+\dfrac{n}{(1-n)T_{0}}\bigl{[}F_{\text{E}}(T_{\text{m}})-F_{\text{H}}(T_{\text{m}})\bigr{]},&n\geq
n_{c},\\\ S_{n},&n<n_{c},\end{cases}$ (21)

where $S_{n}$ is the Rényi entropies without the zeroth-order phase transition
(19) and $n_{c}$ is the critical parameter of the Rényi entropies derived from
(14)

$n_{\text{m}}=\frac{T_{0}}{T_{\text{m}}}=\frac{1+\alpha^{2}}{\sqrt{(3-\alpha^{2})(3\alpha^{2}-1)}}\geq
1.$ (22)

From (20), we have $\lim_{n\to 1}{\mathcal{S}_{\pm}(\alpha,n)}=1$, so the
$n\rightarrow 1$ limit of the Rényi entropies (for any $\alpha$) gives

$S_{EE}=\frac{L^{2}}{4G}V_{\Sigma}\,,$ (23)

which means that the scalar field does not affect the entanglement entropy
($n=1$), while it affects the Rényi entropies with $n>1$. Besides, the Rényi
entropies are also proposed to satisfy an area law Dong:2016fnf

$n^{2}\partial_{n}\Bigl{(}\frac{n-1}{n}S_{n}\Bigr{)}=\frac{\text{Area}(\text{Brane}_{n})}{4G}=\frac{L^{2}}{4G}\mathcal{S}_{\pm}(\alpha,n)^{\frac{2(1-\alpha^{2})}{1+\alpha^{2}}}V_{\Sigma}\,,$
(24)

where the $\text{Brane}_{n}$ is a bulk codimension-2 cosmic brane homologous
to the entangling region A333Here A is the region enclosed by the spherical
entangling surface., and the brane tension is $T_{n}=(n-1)/4nG$. Since
$\lim_{n\to 1}{\mathcal{S}_{\pm}(\alpha,n)}=1$ and $\lim_{n\to 1}{T_{n}}=0$,
the one-parameter generalized area law will reduce to the RT formula in the
$n\to 1$ limit.

The Rényi entropies as a function of $n$ for different values $\alpha$ are
plotted in figure 2. The most notable cases are when $\alpha$ takes values
corresponding to special cases of STU supergravities444There are U(1)4 gauged
fields in STU supergravities for AdS4 spacetime Cvetic:1999xp , with four
U(1)4 charges, respectively. Here the special values of $\alpha$ correspond to
the cases when some of the charges are the same while others are zero (see
Ren:2019lgw  for more details).. We use the same terminology as in Ren:2019lgw
:

  * •

1-charge black hole ($\alpha=\sqrt{3}$)
This black hole is exactly the MTZ black hole. The free energy is

$F=-\frac{L}{16\pi G}(1+4\pi^{2}L^{2}T^{2})V_{\Sigma}\,.$ (25)

From (5), the Rényi entropies are

$S_{n}=\frac{L^{2}}{8G}\Bigl{(}1+\frac{1}{n}\Bigr{)}V_{\Sigma}$ (26)

for $n\geq 1$. Note that the result has the same $n$-dependence as the
universal result in 2D CFT with a single interval Calabrese:2004eu

$S_{n}(D=2)=\frac{c}{6}\Bigl{(}1+\frac{1}{n}\Bigr{)}\log\frac{l}{\delta}\,,$
(27)

where $c$ is the central charge, $\delta$ is a short-distance cut-off, and $l$
is the size of the interval. Besides, (26) or (27) can also be derived from
the SAdS3 hyperbolic black hole Hung:2011nu .

  * •

2-charge black hole ($\alpha=1$)
In this case, the free energy is linear with the temperature

$F=-\frac{L^{2}}{4G}TV_{\Sigma}\,,$ (28)

which is always less stable than the SAdS black hole. So the Rényi entropies
is

$S_{n}=S_{n}(\alpha=0).$ (29)

  * •

3-charge black hole ($\alpha=1/\sqrt{3}$)
In the nontrivial neutral limit, the system has the invariance under
$\alpha\rightarrow 1/\alpha$. So the free energy and Rényi entropies are the
same as (25) and (26), respectively.

  * •

4-charge black hole ($\alpha=0$)
The black hole is the SAdS black hole. The free energy is

$F=-\frac{L}{108\pi G}\bigl{[}\pi
LT(9+8\pi^{2}L^{2}T^{2})+(3+4\pi^{2}L^{2}T^{2})^{3/2}\bigr{]}V_{\Sigma}\,.$
(30)

The Rényi entropies are

$S_{n}(\alpha=0)=\frac{nL^{2}}{8(n-1)}\biggl{[}2-\frac{1+\sqrt{1+3n^{2}}}{3n}-\biggl{(}\frac{1+\sqrt{1+3n^{2}}}{3n}\biggr{)}^{3}\biggr{]}V_{\Sigma}\,,$
(31)

which exactly gives the result from SAdS4 black hole in Hung:2011nu .

Figure 2: The Rényi entropies as a function of $n$ for different values of
$\alpha$ (AdS4). The dashed line is for the SAdS4 case, and the blue line is
for $\alpha=1/\sqrt{3}$ or $\sqrt{3}$. A third-order phase transition occurs
at $n=1$ for all $\alpha\neq 1$. When $1/\sqrt{3}<\alpha<\sqrt{3}$, a zeroth-
order phase transition of black holes leads to discontinuity of the first
order derivative of Rényi entropies at $n_{c}\geq 1$, marked by a dot.

###  2.3 Some Rényi entropy inequalities

The Rényi entropies should satisfy the following inequalities Beck:1993

$\displaystyle\frac{\partial S_{n}}{\partial n}\leq 0,$ (32)
$\displaystyle\frac{\partial}{\partial
n}\Bigl{(}\frac{n-1}{n}S_{n}\Bigr{)}\geq 0,$ (33)
$\displaystyle\frac{\partial}{\partial n}\bigl{(}(n-1)S_{n}\bigr{)}\geq 0,$
(34) $\displaystyle\frac{\partial^{2}}{\partial
n^{2}}\bigl{(}(n-1)S_{n}\bigr{)}\leq 0.$ (35)

From the holographic calculation, these inequalities are closely related to
the stability of hyperbolic black holes (or the CFT living on the hyperbolic
cylinder), which has been discussed555These inequalities can also be proved
under the assumption of bulk stability against perturbation Nakaguchi:2016zqi
through the holographic description of the Rényi entropies in Dong:2016fnf .
in detail in Hung:2011nu  by combining these inequalities with (6). Among
them, (32) and (35) are ensured by the positive specific heat of black holes,
while (33) is guaranteed by the positivity of thermal entropy. Besides, (34)
is related to both specific heat and thermal entropy.

We can verify that all these inequalities are satisfied for the Renyi
entropies that we obtained from the thermodynamically preferred branch of
black hole solutions. The thermal entropy is positive from (15) since
$r_{h}>0$. The specific heat can be derived from (13) and (15) as

$c=T\frac{\partial S}{\partial
T}=\pm\frac{V_{\Sigma}}{G}\frac{\pi(1-\alpha^{2})L^{3}T}{\sqrt{(\alpha^{2}-3)(3\alpha^{2}-1)+4\pi^{2}(1+\alpha^{2})^{2}L^{2}T^{2}}}\Bigl{(}\frac{r_{h,\pm}}{L}\Bigr{)}^{\frac{2(1-\alpha^{2})}{1-3\alpha^{2}}}.$
(36)

As discussed in section 2.1, the thermodynamically preferred solution is
either the SAdS black hole or one of the hairy branches, which depends on the
values of $\alpha$ and the Rényi parameter $n$. The specific heat of the SAdS
solution is obvious to see by taking $\alpha=0$ ($r_{h,+}$). Between the two
hairy black hole branches, the one with $r_{h,+}$ is more stable when
$0<\alpha<1$, while the one $r_{h,-}$ is more stable when $\alpha>1$. Thus the
specific heat is always positive for the more stable hairy solution. When
$\alpha=1$, the hairy black hole is unstable with zero specific heat, so the
black hole is always SAdS. Therefore, these Rényi entropy inequalities hold
for all values of $\alpha$ (except for the critical points where the
derivatives of the Rényi entropies are discontinuous).

##  3 Holographic Rényi entropies from hyperbolic AdS5 black holes

It is natural to generalize the AdS4 cases to higher dimensions, especially
AdS5, whose special cases also correspond to supergravities. A class of black
hole solutions to the EMD system in higher dimensions were found in Gao:2004tv
; Gao:2005xv . We still focus on the nontrivial neutral limit Ren:2019lgw .

###  3.1 Neutral hyperbolic black holes with scalar hair in AdS5 spacetime

The action in AdS5 is

$S=\int
d^{5}x\sqrt{-g}\Bigl{[}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\Bigr{]},$ (37)

with the potential666The expansion behavior of the potential near
$\phi\rightarrow 0$ is
$V(\phi)=-12/L^{2}-2\phi^{2}/L^{2}+\mathcal{O}(\phi^{3})$, so the mass of the
scalar field is $m^{2}L^{2}=-4$ and the scaling dimension of the dual operator
is $\Delta_{\pm}=2$. The scalar field behaves as
$\phi(r)\sim\alpha\ln{r}/r^{2}+\beta/r^{2}$ near the AdS boundary. The
boundary condition for the following solution corresponds to the standard
quantization $\alpha=0$.

$V(\phi)=-\frac{12}{(4+3\alpha^{2})^{2}L^{2}}\Bigl{[}3\alpha^{2}(3\alpha^{2}-2)e^{-\frac{4\phi}{3\alpha}}+36\alpha^{2}e^{\frac{(3\alpha^{2}-4)\phi}{6\alpha}}+2(8-3\alpha^{2})e^{\alpha\phi}\Bigr{]}.$
(38)

The solution is Ren:2019lgw

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+U(r)d\Sigma_{3}^{2}\ ,\qquad
e^{\alpha\phi}=\biggl{(}1-\frac{b^{2}}{r^{2}}\biggr{)}^{\frac{6\alpha^{2}}{4+3\alpha^{2}}},$
(39)

with

$\displaystyle\begin{split}f(r)&=-\biggl{(}1-\frac{b^{2}}{r^{2}}\biggr{)}^{\frac{4-3\alpha^{2}}{4+3\alpha^{2}}}+\frac{r^{2}}{L^{2}}\biggl{(}1-\frac{b^{2}}{r^{2}}\biggr{)}^{\frac{3\alpha^{2}}{4+3\alpha^{2}}},\\\
g(r)&=f(r)\biggl{(}1-\frac{b^{2}}{r^{2}}\biggr{)}^{\frac{3\alpha^{2}}{4+3\alpha^{2}}},\qquad
U(r)=r^{2}\biggl{(}1-\frac{b^{2}}{r^{2}}\biggr{)}^{\frac{3\alpha^{2}}{4+3\alpha^{2}}}.\end{split}$
(40)

From $f(r_{h})=0$, $b$ can be replaced with

$b^{2}=r_{h}^{2}\Bigl{[}1-\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{4+3\alpha^{2}}{2-3\alpha^{2}}}\Bigr{]},$
(41)

and thus the temperature is

$T=\frac{\sqrt{f^{\prime}(r_{h})g^{\prime}(r_{h})}}{4\pi}=\frac{1}{2\pi(4+3\alpha^{2})L}\biggl{[}(8-3\alpha^{2})\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{4+3\alpha^{2}}{4-6\alpha^{2}}}-(4-6\alpha^{2})\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{-\frac{4+3\alpha^{2}}{4-6\alpha^{2}}}\biggr{]}.$
(42)

So we have

$r_{h,\pm}=\Biggl{(}\frac{\pi
L(3\alpha^{2}+4)T\pm\sqrt{(8-3\alpha^{2})(4-6\alpha^{2})+\pi^{2}L^{2}(3\alpha^{2}+4)^{2}T^{2}}}{8-3\alpha^{2}}\Biggr{)}^{\frac{4-6\alpha^{2}}{4+3\alpha^{2}}}L.$
(43)

Other thermodynamic quantities are

$\displaystyle M$ $\displaystyle=-\frac{3V_{\Sigma}}{16\pi
G}\frac{4-3\alpha^{2}}{4+3\alpha^{2}}r_{h}^{2}\Bigl{[}1-\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{4+3\alpha^{2}}{2-3\alpha^{2}}}\Bigr{]},$
(44) $\displaystyle S$
$\displaystyle=\frac{V_{\Sigma}}{4G}U(r_{h})^{3/2}=\frac{V_{\Sigma}}{4G}\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{3(4-3\alpha^{2})}{2(2-3\alpha^{2})}}L^{3},$
(45) $\displaystyle F$ $\displaystyle=M-TS=-\frac{V_{\Sigma}}{16\pi
G}r_{h}^{2}\biggl{[}1+\Bigl{(}\frac{r_{h}}{L}\Bigr{)}^{\frac{4+3\alpha^{2}}{2-3\alpha^{2}}}\biggr{]},$
(46)

from which we can verify the phase transitions and the Rényi entropy
inequalities as in section 2.

Similar to the discussion in AdS4 cases, (43) shows that there may be two
solutions of black holes at a given temperature: $r_{h,+}$ is real when
$0\leq\alpha<4/\sqrt{6}$, while $r_{h,-}$ is real when $\alpha>2/\sqrt{6}$.
The temperature can reach zero when $0\leq\alpha\leq 2/\sqrt{6}$ or
$\alpha\geq 4/\sqrt{6}$, while there is a minimum temperature when
$2/\sqrt{6}<\alpha<4/\sqrt{6}$:

$T_{\text{m}}=\frac{\sqrt{2(3\alpha^{2}-2)(8-3\alpha^{2})}}{\pi(3\alpha^{2}+4)L}\leq
T_{0},$ (47)

where a zeroth-order phase transition from hairy black holes to the SAdS black
hole occurs. The maximum value of $T_{\text{m}}$ is $T_{0}=1/2\pi L$ in the
case $\alpha=2/\sqrt{3}$. Besides, there is a third-order phase transition
from the SAdS hole to hairy black holes at $T=T_{0}$ for any $\alpha\neq
2/\sqrt{3}$.

###  3.2 Holographic Rényi entropies

The calculation of the Rényi entropies is similar to that in section 2. We
directly write the general cases without the zeroth-order phase transition
($0\leq\alpha\leq 2/\sqrt{6}$ and $\alpha\geq 4/\sqrt{6}$)

$S_{n}(n\geq
1)=\frac{nL^{3}}{8(n-1)G}\Bigl{[}2-\mathcal{S}_{\pm}(\alpha,n)^{\frac{8-12\alpha^{2}}{4+3\alpha^{2}}}-\mathcal{S}_{\pm}(\alpha,n)^{\frac{16-6\alpha^{2}}{4+3\alpha^{2}}}\Bigr{]}V_{\Sigma}\,,$
(48)

with

$\mathcal{S}_{\pm}(\alpha,n)=\frac{1}{2(8-3\alpha^{2})n}\Bigl{[}(4+3\alpha^{2})\pm\sqrt{(4+3\alpha^{2})^{2}+8(3\alpha^{2}-8)(3\alpha^{2}-2)n^{2}}\Bigr{]},$
(49)

where the sign corresponds to $r_{h,\pm}$ as in AdS4 cases. When the system
experiences the zeroth-order phase transition, we can still use (21) to
express the Rényi entropies for $2/\sqrt{6}<\alpha<4/\sqrt{6}$, with the
critical parameter

$n_{\text{m}}=\frac{T_{0}}{T_{\text{m}}}=\frac{4+3\alpha^{2}}{\sqrt{2(3\alpha^{2}-2)(8-3\alpha^{2})}}.$
(50)

The black holes corresponding to special cases of supergravities are777Unlike
the AdS4 cases with the four U(1) gauge fields, there are three U(1) gauge
fields in AdS5 in STU supergravities Cvetic:1999xp , and thus we have three
U(1) charges here.:

  * •

1-charge black hole ($\alpha=4/\sqrt{6}$)
We find the free energy when $\alpha=4/\sqrt{6}$ in AdS5 is exactly the same
as the one when $\alpha=\sqrt{3}$ in AdS4. The results are

$\displaystyle F=-\frac{L^{2}}{16\pi
G}(1+4\pi^{2}L^{2}T^{2})V_{\Sigma}\,,\qquad
S_{n}=\frac{L^{3}}{8G}\Bigl{(}1+\frac{1}{n}\Bigr{)}V_{\Sigma}\,.$ (51)

  * •

2-charge black hole ($\alpha=2/\sqrt{6}$)
This solution corresponds to the same system with the $1$-charge black hole
(51).

  * •

3-charge black hole ($\alpha=0$)
The scalar hair vanishes, and we obtain the SAdS5 black hole with the free
energy

$F=-\frac{L^{2}}{128\pi G}\Bigl{[}6+4\pi LT\Bigl{(}\pi
LT(3+\pi^{2}L^{2}T^{2})+(2+\pi^{2}L^{2}T^{2})^{3/2}\Bigr{)}\Bigr{]}V_{\Sigma}\,.$
(52)

And the Rényi entropies are

$S_{n}(\alpha=0)=\frac{nL^{3}}{8(n-1)G}\biggl{[}2-\biggl{(}\frac{1+\sqrt{1+8n^{2}}}{4n}\biggr{)}^{2}-\biggl{(}\frac{1+\sqrt{1+8n^{2}}}{4n}\biggr{)}^{4}\biggr{]}V_{\Sigma}\,.$
(53)

Figure 3: The Rényi entropies as a function of $n$ for different values of
$\alpha$ (AdS5). The dashed line is for the SAdS5 case, and the blue line is
for $\alpha=2/\sqrt{6}$ or $4/\sqrt{6}$. A third-order phase transition occurs
at $n=1$ for all $\alpha\neq 2/\sqrt{3}$. When $2/\sqrt{6}<\alpha<4/\sqrt{6}$,
a zeroth-order phase transition of black holes leads to discontinuity of the
first order derivative of Rényi entropies at $n_{c}\geq 1$, marked by a dot.

##  4 Entanglement spectrum

The entanglement spectrum is the eigenvalue distribution of the reduced
density matrix. With the Rényi entropies of all orders, the entanglement
spectrum is determined. Generally, the spectrum should include both discrete
and continuous parts Hung:2011nu . We briefly review the analysis. Assuming
that the eigenvalue distribution is discrete, we can write the Rényi entropies
as

$S_{n}=\frac{1}{1-n}\log\text{Tr}[\rho^{n}]=\frac{1}{1-n}\log\Bigl{(}\sum_{i}{d_{i}\lambda_{i}^{n}}\Bigr{)},$
(54)

where $\lambda_{i}$ is the eigenvalue of the reduced density matrix which
satisfies $\lambda_{1}>\lambda_{2}>\cdots$, and $d_{i}$ is the degeneracy of
$\lambda_{i}$. The large $n$ expansion of (54) gives

$S_{n}=-\log\lambda_{1}-\frac{1}{n}\log(d_{1}\lambda_{1})+\mathcal{O}\Bigl{[}\frac{1}{n^{2}},\
\frac{1}{n}\Bigl{(}\frac{\lambda_{2}}{\lambda_{1}}\Bigr{)}^{n}\Bigr{]},$ (55)

where the last term leads to non-analyticity in $1/n$. A continuous spectrum
will also lead to non-analyticity. However, the holographic calculations in
section 2 and 3 do not show any non-analyticity in $1/n$, and thus the
spectrum should include both discrete and continuous parts. Furthermore, from
the thermal perspective of the dual CFT, the analyticity allows only one
discrete eigenvalue $\lambda_{1}$, which is also the largest eigenvalue of the
continuous spectrum Hung:2011nu .

In terms of the analysis above, the Rényi entropies can be written as

$S_{n}=\frac{1}{1-n}\log\biggl{[}d_{1}\lambda_{1}^{n}+\int_{0}^{\lambda_{1}}{\bar{\rho}(\lambda)\lambda^{n}d\lambda}\biggr{]},$
(56)

where $\bar{\rho}(\lambda)$ is the continuous part of the entanglement
spectrum $\rho(\lambda)$. Moreover, $\bar{\rho}(\lambda)$ is analytic at
$\lambda=\lambda_{1}$, and we can write the discrete part in $\rho(\lambda)$
as a Dirac delta function. Thus, the Rényi entropies satisfy

$\exp[(1-n)S_{n}]=\int_{t_{1}}^{+\infty}{e^{-(n+1)t}\rho(e^{-{t}})dt},$ (57)

where $\lambda$ is reparameterized as $e^{-t}$. The lower limit of the
integral (57) can be changed to zero, which gives nothing but the Laplace
transform. Thus, the spectrum can be derived from an inverse Laplace transform
with $n$ as the parameter

$\rho(\lambda)=\frac{1}{\lambda}\mathcal{L}^{-1}\bigl{[}e^{(1-n)S_{n}},n,t\bigr{]}\bigl{|}_{t=-\log\lambda}=\frac{1}{\lambda}\frac{1}{2\pi
i}\lim_{T\rightarrow\infty}\int_{\gamma-
iT}^{\gamma+iT}{e^{(1-n)S_{n}}e^{nt}dn},$ (58)

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.

###  4.1 Entanglement spectrum from AdS4 black holes

We start with special cases when $\alpha=1/\sqrt{3}$ or $\sqrt{3}$, from which
the spectrum can be derived straightforwardly. The Rényi entropies have a
simple form

$S_{n}=\frac{L^{2}}{8G}\Bigl{(}1+\frac{1}{n}\Bigr{)}V_{\Sigma}\,.$ (59)

This result has the same $n$-dependence as the universal result (27) in 2D
CFT, whose entanglement spectrum can be calculated by means of properties of
the polylogarithm function Calabrese:2008cl  or Mellin’s inversion formula
Romero-Bermudez:2018dim :

$\rho(\lambda)=\delta(\lambda_{1}-\lambda)+\frac{b\,\theta(\lambda_{1}-\lambda)}{\lambda\sqrt{b\ln{(\lambda_{1}/\lambda)}}}\,I_{1}\bigl{(}2\sqrt{b\ln{(\lambda_{1}/\lambda)}}\bigr{)},$
(60)

with

$b=-\ln{\lambda_{1}}=\frac{L^{2}}{8G}V_{\Sigma}\,,$ (61)

where $I_{k}(x)$ is the modified Bessel function of the first kind and
$\theta(x)$ is the Heaviside step function.

In general cases, we can still calculate the entanglement spectrum in terms of
an infinite sum if the Rényi entropies are analytic at $n=\infty$. This
analyticity allows us to expand the Rényi entropies near $n=\infty$ as

$S_{n}=\sum\limits_{i=0}^{\infty}{s_{i}n^{-i}}\,,$ (62)

where the coefficients of the first two terms are888Here we discuss the cases
without zeroth-order phase transitions (i.e., $0\leq\alpha\leq 1/\sqrt{3}$ or
$\alpha\geq\sqrt{3}$).

$\displaystyle\begin{split}s_{0}&=\frac{L^{2}}{4G}\Bigl{[}1-2\lvert
1-\alpha^{2}\rvert\cdot\lvert
1-3\alpha^{2}\rvert^{\frac{1-3\alpha^{2}}{2(1+\alpha^{2})}}\lvert
3-\alpha^{2}\rvert^{\frac{\alpha^{2}-3}{2(1+\alpha^{2})}}\Bigr{]}V_{\Sigma}\,,\\\
s_{1}&=s_{0}-\frac{L^{2}}{4G}\Bigl{(}\frac{1-3\alpha^{2}}{3-\alpha^{2}}\Bigr{)}^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}V_{\Sigma}\,,\end{split}$
(63)

whose limits at $\alpha=1/\sqrt{3}$ and $\sqrt{3}$ are

$s_{0}=s_{1}=\frac{L^{2}}{8G}V_{\Sigma}\,.$ (64)

By the approach in Belin:2013dva , the inverse Laplace transform of the LHS of
(57) can be calculated by writing it as an infinite sum. Here we give more
analytic details together with our explicit expression of $S_{n}$. We expand
the LHS of (57) as

$\displaystyle\begin{split}\exp[(1-n)S_{n}]&=\exp(s_{0}-s_{1}-s_{0}n)\exp\biggl{(}\sum\limits_{i=1}^{\infty}{u_{i}n^{-i}}\biggr{)}\\\
&=\biggl{(}1+\sum\limits_{i=1}^{\infty}{v_{i}n^{-i}}\biggr{)}\exp(s_{0}-s_{1}-s_{0}n),\end{split}$
(65)

where $u_{i}$ and $v_{i}$ can be written as the function of expansion
coefficients of $S_{n}$, i.e.,

$u_{i}=s_{i}-s_{i+1},\qquad
v_{i}=\frac{1}{i!}B_{i}(u_{1},\,2!u_{2},\,\ldots,\,i!u_{i}),$ (66)

where $B_{i}$ is the complete exponential Bell polynomial, which can be
expressed as an infinite sum (see appendix B for details).

The inverse Laplace transform of (65) is given by the convolution theorem

$\begin{split}\rho(\lambda)&=\frac{\exp(s_{0}-s_{1})}{\lambda}\biggl{(}\lambda_{1}\delta(\lambda_{1}-\lambda)+\sum_{i=0}^{\infty}{\frac{v_{i+1}}{i!}\bigl{(}\ln(\lambda_{1}/\lambda)\bigr{)}^{i}}\theta(\lambda_{1}-\lambda)\biggr{)}\\\
&=\exp(s_{0}-s_{1})\biggl{(}\delta(\lambda_{1}-\lambda)+\frac{\theta(\lambda_{1}-\lambda)}{\lambda}\sum_{i=0}^{\infty}{\frac{v_{i+1}}{i!}\bigl{(}\ln(\lambda_{1}/\lambda)\bigr{)}^{i}}\biggr{)}.\end{split}$
(67)

Similar to the special case (60), the general spectrum also includes a Dirac
delta function and Heaviside step function determined by the largest
eigenvalue $\lambda_{1}$, which provides an example to verify the analysis
above that the spectrum consists of one discrete eigenvalue and the continuous
part. By comparing (67) and the ansatz (56), the degeneracy of the eigenvalue
$\lambda_{1}$ is

$d_{1}=\exp(s_{0}-s_{1})=\exp\biggl{[}\frac{L^{2}}{4G}\biggl{(}\frac{1-3\alpha^{2}}{3-\alpha^{2}}\biggr{)}^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}V_{\Sigma}\biggr{]},$
(68)

which reduces to $d_{1}=1$ when $\alpha=1/\sqrt{3}$ and $\sqrt{3}$. The
largest eigenvalue of the spectrum is given by the constant term of the Rényi
entropies

$\displaystyle\begin{split}&\lambda_{1}=\exp(-s_{0})\\\
&=\begin{cases}\exp\Bigl{[}\frac{L^{2}}{4G}\bigl{(}2\lvert
1-\alpha^{2}\rvert\cdot\lvert
1-3\alpha^{2}\rvert^{\frac{1-3\alpha^{2}}{2(\alpha^{2}+1)}}\lvert
3-\alpha^{2}\rvert^{\frac{\alpha^{2}-3}{2(\alpha^{2}+1)}}-1\bigr{)}V_{\Sigma}\Bigr{]},&0\leq\alpha\leq
1/\sqrt{3}\ \text{or}\ \alpha\geq\sqrt{3},\\\
\exp\bigl{[}F_{H}(T_{0})-F_{H}(T_{\text{m}})+F_{S}(T_{\text{m}})-F_{S}(0)\bigr{]},\
&1/\sqrt{3}<\alpha<\sqrt{3}.\end{cases}\end{split}$ (69)

where $F_{H}$ and $F_{S}$ are the free energy of hairy and SAdS black holes,
respectively, and $T_{\text{m}}$ is the phase transition temperature given by
(14). The above calculations show that the largest eigenvalue of the
entanglement spectrum decreases while its degeneracy increases if we enlarge
the entangling sphere. Besides, the density of the continuous part
$\bar{\rho}(\lambda)$ increases. The normalization of the spectrum (67) can be
verified numerically to satisfy

$d_{1}\lambda_{1}+\int_{0}^{\lambda_{1}}{\lambda\bar{\rho}(\lambda)d\lambda}=1.$
(70)

As a special example, when $\alpha=1/\sqrt{3}$ or $\sqrt{3}$, we have
$s_{0}=s_{1}$ and $s_{i}=0$ (for $i>1$). Inserting them into (66) and (67), we
obtain the spectrum

$\displaystyle\begin{split}\rho(\lambda)&=\delta(\lambda_{1}-\lambda)+\frac{1}{\lambda}\sum_{i=0}^{\infty}{\frac{s_{0}^{i+1}}{i!(i+1)!}\bigl{(}\ln(\lambda_{1}/\lambda)\bigr{)}^{i}}\theta(\lambda_{1}-\lambda)\\\
&=\delta(\lambda_{1}-\lambda)+\frac{s_{0}\theta(\lambda_{1}-\lambda)}{\lambda\sqrt{s_{0}\ln(\lambda_{1}/\lambda)}}I_{1}\bigl{(}2\sqrt{s_{0}\ln(\lambda_{1}/\lambda)}\bigr{)},\end{split}$
(71)

which exactly gives the result (60). For general cases, the entanglement
spectrum is shown in figure 4 and 5.

Figure 4: The entanglement spectrum as a function of $\lambda$. The dashed
line is for the SAdS4 black hole, while the solid lines are for hairy black
holes, with $\alpha=0.52$ (orange line) and $1/\sqrt{3}$ or $\sqrt{3}$ (blue
line). The largest eigenvalues are 0.857, 0.869, 0.882, respectively. The
largest eigenvalue for different $\alpha$ is shown in the subfigure.

The spectrum in AdS5 is the same as (71) when $\alpha=2/\sqrt{6}$ or
$4/\sqrt{6}$.

We note that the expansion of the Rényi entropies at $n=\infty$ (62) has a
radius of convergence. From (19) and (20), the expansion converges for $n$
larger than a special value $n_{c}$ determined by $\alpha$ ($0\leq\alpha\leq
1/\sqrt{3}$ and $\alpha\geq\sqrt{3}$)

$n_{c}=\frac{1+\alpha^{2}}{\sqrt{(3-\alpha^{2})(1-3\alpha^{2})}},$ (72)

which increases as $\alpha$ approaches $1/\sqrt{3}$. When
$\alpha\leq\sqrt{2}-1$ or $\alpha\geq\sqrt{2}+1$, the expansion can converge
at $n=1$. Otherwise, we shall not expect the expansion (62) to describe the
Rényi entropies of all orders, especially when $\alpha$ is extremely close to
$1/\sqrt{3}$ or $\sqrt{3}$, where different $\alpha$ correspond to
dramatically different systems. This leads to non-commutativity of the
$\alpha\to 1/\sqrt{3}$ and $n\to\infty$ limits of the Rényi entropies.
Consequently, the spectrum given by the series (67) is expected to explicitly
describe the low-lying part999Following Haldane:2008en , we use the term low-
lying, which represents the spectrum with lower energy, or with $\lambda$
close to the largest eigenvalue. (larger $\lambda$), which is determined by
the large $n$ behavior of the Rényi entropies, while the spectrum for
$\lambda$ close to zero may be inconsistent with (67).

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

$\frac{\partial}{\partial
n}\bigl{(}(n-1)S_{n}\bigr{)}\Bigl{|}_{n_{0}}+\ln{\lambda}=0,$ (73)

and the integral (58) is approximated as

$\rho(\lambda)=\frac{1}{\lambda^{n+1}}e^{(1-n)S_{n}}\Bigl{[}2\pi\frac{\partial^{2}}{\partial
n^{2}}\bigl{(}(1-n)S_{n}\bigr{)}\Bigr{]}^{-1/2}\Big{|}_{n_{0}}.$ (74)

For instance, the saddle point is
$n_{0}=\sqrt{s_{0}/\ln(\lambda_{1}/\lambda)}$ when $\alpha=1/\sqrt{3}$. The
consistency is convenient to verify by changing the variable $t=-\ln\lambda$

$\rho(t\to\infty)\sim\frac{1}{16\sqrt{\pi}}\bigl{[}s_{0}(t-s_{0})\bigr{]}^{-3/4}e^{2\sqrt{s_{0}(t-s_{0})}},$
(75)

which gives the leading term of the asymptotic expansion of (71) at
$t=\infty$. Considering the phase transition at $n=1$, all these spectra
should have the same $\lambda\to 0$ behavior (as the SAdS solution), whose
saddle point is $n_{0}=(2/t)^{1/3}/3+\mathcal{O}(t^{-4/3})$.

Figure 5: The entanglement spectrum near $\alpha=1/\sqrt{3}$. The dashed line
is for the SAdS4 black hole, while the solid lines are for hairy black holes,
where the yellow, green, brown, purple, red, and blue lines are for
$\alpha=0.5,\ 0.576,\ 0.577,\ 0.5773,\ 0.57735$, and $1/\sqrt{3}$,
respectively. Here we plot only the continuous spectrum.

As a comparison, we also employ numerical methods101010Some details of
numerical methods are given in Cheng:1998IO  (or see appendix B). Different
methods can give different results, some of which are unreliable in our cases.
Because of the great change when $\alpha$ approaches $1/\sqrt{3}$, a
singularity of (19), we test the accuracy of numerical methods by comparing
the cases when $\alpha$ is extremely close to $1/\sqrt{3}$ with (60), shown in
figure 5. to evaluate the inverse Laplace transform, which shows the spectrum
obtained from (67) is accurate for a wide range of $\lambda$ as long as
$\alpha$ is not extremely close to $1/\sqrt{3}$. Therefore, the series (67) is
sufficient to describe the low-lying entanglement spectrum, except when
$\alpha$ is extremely close to $1/\sqrt{3}$. The low-lying spectrum for
$\alpha$ close to $1/\sqrt{3}$ is plotted in figure 5 numerically, from which
we can see the spectrum has unique behavior when $\lambda$ approaches
$\lambda_{1}$:

  * •

When $\alpha$ is not extremely close to $1/\sqrt{3}$, the spectrum changes
with respect to $\alpha$ as a whole, and the largest eigenvalue increases, as
expected.

  * •

When $\alpha$ is extremely close to $1/\sqrt{3}$, the continuous part of the
spectrum near $\lambda_{1}$ experiences dramatical change: the spectral
density of large eigenvalues increases with $\alpha$, while the spectrum with
small eigenvalues is almost fixed. The degeneracy of $\lambda_{1}$ decreases
while the location keeps almost fixed.

In the above calculations, we neglect the third-order phase transition of the
Rényi entropies because the spectrum of the low-lying part is not affected by
the phase transition at $n=1$, based on the following observations. First, the
spectrum near the largest eigenvalue $\lambda_{1}$ is entirely determined by
the large $n$ behavior of the Rényi entropies. Second, the saddle point
approximation shows that the spectrum with $\lambda\to 0$ is dominated by the
Rényi entropies with $n\to 0$. Furthermore, when we perform the expansion (62)
and (65), we find the number of terms and precision of the expansion would
significantly affect the Rényi entropies (62) beyond the convergent region.
However, the spectrum obtained from these expansions does not show any
difference for a wide range of $\lambda$. Another similar observation comes
from the comparison between some numerical methods and (67), which shows
consistency until $\lambda$ is small enough (near zero). These observations
suggest the spectrum of different $\lambda$ is affected by different parts of
the Rényi entropies: the spectrum with large $\lambda$ is not significantly
affected by small $n$. As a consequence, the phase transition at $n=1$ is
merely embodied in the spectrum with $\lambda$ close to zero, and the spectra
for different $\alpha$ have the same asymptotic behavior (as the SAdS black
hole) at $\lambda\to 0$.

In condensed matter literature, people are more interested in the low-lying
part of the entanglement spectrum, which implies more universal properties of
the system Haldane:2008en ; Calabrese:2008cl . Especially in two dimensions,
the strong conformal invariance fixes the continuous spectrum entirely, so the
universal information beyond the entanglement entropy is embodied in the
location and degeneracies of the low-lying eigenvalues Calabrese:2008cl .
However, the cases in higher dimensions imply more complex properties. The
systems corresponding to different $\alpha$ have different continuous spectra
as well as discrete parts.

##  5 Summary and Discussion

In this paper, we have calculated the Rényi entropies (with a spherical
entangling surface) and entanglement spectrum from a class of hyperbolic black
holes with scalar hair in terms of the conformal mapping approach
Casini:2011kv ; Hung:2011nu . The main conclusions are as follows:

  * •

By employing a class of hyperbolic black holes with scalar hair as a one-
parameter family generalization of the MTZ black hole, we explicitly obtain
the holographic Renyi entropies. The $n\to 1$ limit of these Rényi entropies
gives the same entanglement entropy, while the Rényi entropies with $n>1$ are
affected by the scalar field. In special cases when $\alpha=1/\sqrt{3}$ or
$\sqrt{3}$ (in AdS4) and $\alpha=2/\sqrt{6}$ or $4/\sqrt{6}$ (in AdS5), which
correspond to special cases of STU supergravities, the Rényi entropies have
the same $n$-dependence as the universal result in 2D CFT.

  * •

The zeroth-order and third-order phase transitions of black holes lead to the
discontinuity in the first and second derivatives of the Rényi entropies at
special values $n_{\text{m}}$, which depend on the phase transition
temperature of black holes.

  * •

From the Rényi entropies which are analytic at $n=\infty$, we calculate the
entanglement spectrum as an infinite sum by means of the exponential Bell
polynomial. The result includes a Dirac delta function and a Heaviside step
function determined by the large $n$ limit of the Rényi entropies. This
approach is verified to be accurate for a wide range of eigenvalues by
comparing it with numerical calculations. We take a closer look at the
spectrum when $\alpha$ is extremely close to $1/\sqrt{3}$ (in AdS4), which
shows different properties.

Some interesting questions are waiting for further investigation. In the
discussion above, we leave aside the cases with the zeroth-order phase
transitions due to the difficulty of inverting the Laplace transform. However,
it would be intriguing to explicitly show how phase transitions are embodied
in the entanglement spectrum. Besides, this work focuses on the neutral limit,
but the charged hyperbolic black holes include more complex phase structures
and are worth investigating.

###### Acknowledgements.

We thank Yi Wang for helpful conversations. This work was supported in part by
the NSF of China under Grant No. 11905298 and the Undergraduate Base
Scientific Research Project under Grant No. 20211222.

##  Appendix A Black hole thermodynamics with generalized boundary condition

Since the thermodynamics of the hyperbolic black holes plays a key role in
calculating the Rényi entropies, we study the holographic renormalization of
the black holes with scalar hair by giving the boundary counter terms and
generalized boundary conditions.

The free energy of a black hole is calculated from the renormalized on-shell
action with proper boundary counter terms. For our Einstein-scalar system in
AdS4 with $m^{2}L^{2}=-2$, we consider the following boundary terms111111We
obtain exactly the same renormalized on-shell action as in Caldarelli:2016nni
; Faulkner:2010gj , although the expressions of the boundary terms are
slightly different.

$S_{\partial}=\frac{1}{16\pi G}\int
d^{3}x\sqrt{-\gamma}\biggl{(}2K-\frac{4}{L}-LR[\gamma]-\frac{1}{2L}\phi^{2}-\frac{W(\phi_{a})}{L\phi_{a}^{3}}\phi^{3}\biggr{)},$
(76)

where $\gamma_{\mu\nu}$ is the induced metric, $K$ is the extrinsic curvature
calculated from the unit normal vector pointing out of the AdS boundary, and
$\phi_{a}$ is a coefficient in (89) below. The first term being the Gibbons-
Hawking boundary term for a well-defined Dirichlet variational problem and the
following three terms remove the divergence in the bulk action. The last term
of (76) is a finite boundary term Anabalon:2015xvl . We set the AdS radius
$L=1$ in the following. By variation of the action with respect to the induced
metric $\gamma_{\mu\nu}$, we obtain the boundary stress tensor

$\langle T_{\mu\nu}\rangle=\frac{1}{16\pi G}\lim_{z\to
0}z^{-1}\biggl{[}2(Kh_{\mu\nu}-K_{\mu\nu}-2h_{\mu\nu}+G_{\mu\nu})-\biggl{(}\frac{1}{2}\phi^{2}+\frac{W(\phi_{a})}{\phi_{a}^{3}}\phi^{3}\biggr{)}h_{\mu\nu}\biggr{]},$
(77)

where $G_{\mu\nu}$ is the Einstein tensor calculated from $\gamma_{\mu\nu}$.

We formulate the first law of thermodynamics without imposing a boundary
condition, and then show that the first law of thermodynamics is unmodified by
a scalar charge with a generalized boundary condition
$\phi_{b}=W^{\prime}(\phi_{a})$, whose special cases correspond to multi-trace
deformations in the dual CFT. We will follow the procedure in Li:2020spf  with
a generalization for imposing boundary conditions of the scalar field. The
choice $W(\phi_{a})=c\phi_{a}^{3}$ ($c$ is a constant) gives the same result
as in Li:2020spf .

For convenience of numerical calculations, and to see the difference between
planar and hyperbolic/spherical black holes, we use the metric ansatz

$ds^{2}=\frac{1}{z^{2}}\biggl{(}-g(z)e^{-\chi(z)}dt^{2}+\frac{dz^{2}}{g(z)}+d\Sigma_{2,k}^{2}\biggr{)},$
(78)

where $k=0,-1,1$ are for planar, hyperbolic, and spherical black holes,
respectively. In practice, we can write $d\Sigma_{2,k}^{2}$ as

$d\Sigma_{2,k}^{2}=\frac{dx^{2}}{1-kx^{2}}+(1-kx^{2})d\varphi^{2}\,.$ (79)

The AdS boundary is at $z=0$, and the horizon is at $z=z_{h}$. Unlike the
$k=0$ case in which $z_{h}$ can be fixed to be $z_{h}=1$ by scaling
symmetries, $z_{h}$ is an extra parameter when $k\neq 0$. The equations of
motion for the metric and the scalar field are

$\displaystyle
g^{\prime}-\left(\frac{\chi^{\prime}}{2}+\frac{3}{z}\right)g-\frac{1}{2z}V(\phi)+kz=0\,,$
(80) $\displaystyle\chi^{\prime}-\frac{1}{2}z\phi^{\prime 2}=0\,,$ (81)
$\displaystyle\phi^{\prime\prime}+\left(\frac{g^{\prime}}{g}-\frac{\chi^{\prime}}{2}-\frac{2}{z}\right)\phi-\frac{1}{z^{2}g}V^{\prime}(\phi)=0\,.$
(82)

Near the horizon $z=z_{h}$, the asymptotic behavior of the functions is

$\displaystyle g$
$\displaystyle=\bar{g}_{1}(z_{h}-z)+\bar{g}_{2}(z_{h}-z)^{2}+\cdots,$ (83)
$\displaystyle\chi$ $\displaystyle=\chi_{h}+\bar{\chi}_{1}(z_{h}-z)+\cdots,$
(84) $\displaystyle\phi$
$\displaystyle=\phi_{h}+\bar{\phi}_{1}(z_{h}-z)+\cdots,$ (85)

where $\chi_{h}$ and $\phi_{h}$ can be used to express other coefficients. The
temperature and entropy density are given by

$T=\frac{\bar{g}_{1}e^{-\chi_{h}/2}}{4\pi},\qquad s=\frac{1}{4Gz_{h}^{2}}\,.$
(86)

Near the AdS boundary $z=0$, the asymptotic behavior of the functions is

$\displaystyle g$
$\displaystyle=1+\Bigl{(}k+\frac{1}{4}\phi_{a}^{2}\Bigr{)}z^{2}+g_{3}z^{3}+\cdots,$
(87) $\displaystyle e^{-\chi}g$
$\displaystyle=1+kz^{2}+\Bigl{(}g_{3}-\frac{2}{3}\phi_{a}\phi_{b}\Bigr{)}z^{3}+\cdots,$
(88) $\displaystyle\phi$ $\displaystyle=\phi_{a}z+\phi_{b}z^{2}+\cdots,$ (89)

where higher-order coefficients generally depend on the specific potential
$V(\phi)$. By substituting the boundary expansions into (77), we obtain the
stress tensor. Let $\langle
T^{\mu}_{\;\,\nu}\rangle=\text{diag}(-\varepsilon,\,p,\,p,\,p)$, and we have

$\displaystyle\varepsilon$ $\displaystyle=\frac{1}{16\pi
G}\bigl{(}-2g_{3}+\phi_{a}\phi_{b}+W(\phi_{a})\bigr{)},$ (90) $\displaystyle
p$ $\displaystyle=\frac{1}{16\pi
G}\bigl{(}-g_{3}+\phi_{a}\phi_{b}-W(\phi_{a})\bigr{)},$ (91)

where $\varepsilon$ is the energy density and $p$ is the pressure density. The
trace of the stress tensor is

$\langle T^{\mu}_{\;\,\mu}\rangle=-\varepsilon+2p=\frac{1}{16\pi
G}\bigl{(}\phi_{a}\phi_{b}-3W(\phi_{a})\bigr{)}.$ (92)

We adopt a general procedure developed by Wald Wald:1993nt  to formulate the
first law of black hole thermodynamics. By variation of parameters in a
$(d+1)$-dimensional solution, there is a closed $(d-1)$-form $\delta
Q-i_{\xi}\Theta$ with $\xi$ being a Killing vector. Here we take
$\xi=\partial/\partial_{t}$. Thus, the variation of a Hamiltonian is defined
by the integral

$\delta H=\int_{\Sigma^{(d-1)}}(\delta Q-i_{\xi}\Theta)\,,$ (93)

where $\Sigma^{(d-1)}$ is a $(d-1)$-dimensional surface at constant $t$ and
$z$. A key observation in Wald:1993nt  is that $\delta H$ is a radially
conserved quantity, i.e., $\delta H$ is independent of $z$, and takes the same
value at the horizon and the AdS boundary. For the construction of $\delta H$
in Einstein-scalar systems, see Liu:2013gja ; Lu:2014maa .

For the Einstein-scalar system (7) with the metric (78), $\delta H$ at radius
$z$ is given by

$\delta H=-\frac{V_{\Sigma}}{16\pi
G}z^{-2}e^{-\chi/2}\biggl{(}\frac{2}{z}\delta
g-g\phi^{\prime}\delta\phi\biggr{)},$ (94)

where $V_{\Sigma}$ is the volume of $d\Sigma_{2,k}^{2}$. First we evaluate
$\delta H$ at the horizon. From (83), we have $\delta
g|_{z=z_{h}}=\bar{g}_{1}\delta z_{h}$, and we obtain

$\frac{\delta H}{V_{\Sigma}}\Bigr{|}_{z=z_{h}}=-\frac{1}{8\pi
G}e^{-\chi(z_{h})/2}\bar{g}_{1}z_{h}^{-3}\delta z_{h}=T\delta s\,,$ (95)

where we have used $\delta s=\delta(1/(4Gz_{h}^{2}))=-1/(2Gz_{h}^{3})\delta
z_{h}$. Evaluating $\delta H$ at the AdS boundary gives

$\frac{\delta H}{V_{\Sigma}}\Bigr{|}_{z=0}=-\frac{1}{16\pi G}\lim_{z\to
0}z^{-2}e^{-\chi/2}\biggl{(}\frac{2}{z}\delta
g-g\phi^{\prime}\delta\phi\biggr{)}=\frac{1}{16\pi G}(-2\delta
g_{3}+\phi_{a}\delta\phi_{b}+2\phi_{b}\delta\phi_{a})\,.$ (96)

The variation of the energy density (90) gives

$\delta\varepsilon=\frac{1}{16\pi G}\bigl{(}-2\delta
g_{3}+\phi_{a}\delta\phi_{b}+(\phi_{b}+W^{\prime}(\phi_{a}))\delta\phi_{a}\bigr{)}.$
(97)

Then we obtain

$\frac{\delta H}{V_{\Sigma}}\Bigr{|}_{z=0}=\delta\varepsilon+\frac{1}{16\pi
G}(\phi_{b}-W^{\prime}(\phi_{a}))\delta\phi_{a}\,,$ (98)

We will set $16\pi G=1$ for simplicity. By equating $\delta H$ at the horizon
and the boundary, we obtain the first law of thermodynamics:

$\delta\varepsilon=T\delta s-(\phi_{b}-W^{\prime}(\phi_{a}))\delta\phi_{a}\,.$
(99)

We can clearly see that upon imposing a generalized boundary condition
$\phi_{b}=W^{\prime}(\phi_{a})$, the first law of thermodynamics is unmodified
by the scalar field.

In other words, for a boundary condition parametrized by
$\phi_{b}=W^{\prime}(\phi_{a})$, we can always formulate the first law of
thermodynamics $d\varepsilon=Tds$ (unmodified by a scalar charge) if we choose
the boundary terms (76). Special cases include multi-trace deformations of the
dual CFT. The double-trace deformation corresponds to
$\phi_{b}/\phi_{a}=\kappa$, and thus $W(\phi_{a})=(1/2)\kappa\phi_{a}^{2}$.
The triple-trace deformation corresponds to $\phi_{b}/\phi_{a}^{2}=\tau$, and
thus $W(\phi_{a})=(1/3)\tau\phi_{a}^{3}$.

The free energy is calculated by the renormalized on-shell action as
$F/T=-(S+S_{\partial})$, where $S$ is the bulk action and $S_{\partial}$ is
the boundary counter terms. The free energy density is the free energy per
volume: $\mathsf{f}=F/V_{\Sigma}$. Evaluating the bulk action gives

$\displaystyle-S$ $\displaystyle=\int
d^{3}x\int_{z_{h}}^{0}dz\biggl{[}\biggl{(}\frac{2}{z^{3}}ge^{-\chi/2}\biggr{)}^{\prime}+2kz^{-2}e^{-\chi/2}\biggr{]}$
(100)
$\displaystyle=TV_{\Sigma}\,\frac{2}{z^{3}}ge^{-\chi/2}\biggr{|}_{z=0}+2kTV_{\Sigma}\int_{z_{h}}^{0}z^{-2}e^{-\chi/2}dz.$
(101)

When $k=0$, the bulk Lagrangian is a total derivative Hartnoll:2008kx , and
thus we can express the action in terms of boundary quantities. When $k\neq
0$, however, the action will inevitably involve an integral from the horizon
to the boundary. Consequently, the free energy density is given by

$\mathsf{f}=\underbrace{g_{3}-\phi_{a}\phi_{b}+W(\phi_{a})}_{\small\begin{array}[]{c}-p\end{array}}+2k\biggl{(}\frac{1}{z}+\int_{z_{h}}^{z}\bar{z}^{-2}e^{-\chi(\bar{z})/2}d\bar{z}\biggr{)}\biggr{|}_{z=0}.$
(102)

We can verify that the thermodynamic relation $\mathsf{f}=\varepsilon-Ts$ is
satisfied by using a radially conserved quantity Cai:2020wrp

$\displaystyle\mathcal{Q}$
$\displaystyle=z^{-2}e^{\chi/2}(ge^{-\chi})^{\prime}+2k\int
z^{-2}e^{-\chi/2}dz$ $\displaystyle\equiv
z^{-2}e^{\chi/2}(ge^{-\chi})^{\prime}+2k\mathcal{P}(z)\,,$ (103)

which satisfies $\mathcal{Q}^{\prime}(z)=0$. Evaluating $\mathcal{Q}$ at the
horizon gives

$\mathcal{Q}|_{z=z_{h}}=-Ts+2k\mathcal{P}(z_{h})\,.$ (104)

Evaluating $\mathcal{Q}$ at the AdS boundary gives

$\mathcal{Q}|_{z=0}=\underbrace{3g_{3}-2\phi_{a}\phi_{b}}_{\small\begin{array}[]{c}-(\varepsilon+p)\end{array}}+2k\biggl{(}\frac{1}{z}+\mathcal{P}(z)\biggr{)}\biggr{|}_{z=0}\,,$
(105)

where we have used (90) and (91). By equating (104) and (105), we obtain the
free energy density $\mathsf{f}=\epsilon-Ts$, which is exactly the same as
(102). We can see that when $k=0$, we have $\mathsf{f}=-p$. For
hyperbolic/spherical black holes, in contrast, the free energy depends on a
bulk integral. Nevertheless, the first law of thermodynamics is valid for all
$k$. We have $d\mathsf{f}=-sdT$ for the boundary condition
$\phi_{b}=W^{\prime}(\phi_{a})$ together with the boundary terms (76).

With the analytic solution in hand, we can verify the first law of
thermodynamics with the boundary condition given by a triple-trace
deformation. For the boundary condition given by a double-trace deformation,
we can numerically solve the Einstein-scalar system (7) with the potential (8)
by the numerical techniques described in Hartnoll:2008kx . Integrating out
from the horizon to the AdS boundary gives a map

$(z_{h},\,\phi_{h})\mapsto(\phi_{a},\,\phi_{b},\,g_{3})\,.$ (106)

Upon imposing the boundary condition $\phi_{b}=\kappa\phi_{a}$, we obtain a
two-parameter family of solutions, where the two parameters are two
dimensionless combinations of ($z_{h}$, $T$, $\kappa$). By extracting the
thermodynamic quantities $\varepsilon$, $T$, and $s$ from the numerical
solution, we have verified that the first law of thermodynamics
$d\varepsilon=Tds$ is satisfied.

##  Appendix B Mathematical notes

###  B.1 Inverse Laplace transform

The Laplace transform of a real variable function $f(t)$ is defined as

$F(s)=\mathcal{L}[f(t)](s)=\int_{0}^{\infty}{f(t)e^{-st}dt}.$ (107)

The inverse Laplace transform can be obtained by Mellin’s inversion formula

$f(t)=\mathcal{L}^{-1}[F(s)](t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-
iT}^{\gamma+iT}{F(s)e^{st}dt}=\sum_{i=1}{\text{Res}[e^{st}F(s),s_{i}]},$ (108)

where the integral is taken over a vertical line with $\text{Re}(s)=\gamma$,
and $\gamma$ is a real number ensuring no singularities on the right side of
this line. Thus the result can be obtained by considering the residues of
$e^{st}F(s)$ at all isolated singularities $s_{i}$ on the complex plane. Two
basic inversions related to our calculations are

$\mathcal{L}^{-1}[e^{-\tau
s}](t)=\delta(t-\tau),\qquad\mathcal{L}^{-1}\Bigl{[}\frac{e^{-\tau
s}}{s}\Bigr{]}(t)=\theta(t-\tau).$ (109)

Generally, the Laplace transform is not easy to invert, and thus we would
apply numerical methods to evaluate the integral (108). The accuracy and
efficiency of different methods were tested and compared in terms of many
analytic results Cheng:1998IO ; Davies:19791 . We make our choice by comparing
the spectrum (with $\alpha\to 1/\sqrt{3}$) obtained from direct numerical
calculations with (60). It turns out that the Stehfest method Stehfest:1970Al
and the Piessens Gaussian Quadrature (PGQ) method Piessens:1971Ga  behave
better in our cases. The Stehfest method approximates the inversion by the
following sum

$f(t)\approx\frac{\ln 2}{t}\sum_{n=1}^{N}c_{n}F\Bigl{(}\frac{n\ln
2}{t}\Bigr{)},$ (110)

with

$c_{n}=(-1)^{n+N/2}\sum_{k=(n+1)/2}^{\min\\{n,N/2\\}}{\frac{k^{N/2}(2k)!}{(N/2-k)!k!(k-1)!(n-k)!(2k-n)!}},$
(111)

where $N$ is an even number. The PGQ method is to evaluate the sum

$f(t)\approx\frac{1}{t}\sum_{i=1}^{N}{w_{i}x_{i}F\Bigl{(}\frac{x_{i}}{t}\Bigr{)}},$
(112)

where $x_{i}$ are zero points of an orthogonal polynomial $P_{i}(x)$ with the
recursive relation

$\begin{split}P_{0}(x)&=1,\qquad P_{1}(x)=x-1,\\\
P_{i}(x)&=\Bigl{[}2(2i-1)x+\frac{2}{2i-3}\Bigr{]}P_{i-1}(x)+\frac{2i-1}{2i-3}P_{i-2}(x).\end{split}$
(113)

And the weight is

$w_{i}=(-1)^{N-1}\frac{1}{N}\biggl{(}\frac{2N-1}{x_{i}P_{N-1}(1/x_{i})}\biggr{)}^{2}.$
(114)

We find that the PGQ method (figure 5) is more accurate when $\lambda$ is
close to the largest eigenvalue $\lambda_{1}$, while the Stehfest method is
accurate when $\lambda$ is not so close to $\lambda_{1}$.

###  B.2 The Bell polynomial

In (66), we use the complete exponential Bell polynomial to express the
expansion coefficients. The complete Bell polynomial has the generating
function

$B_{0}=1,\qquad B_{i}(x_{1},\ldots,x_{i})=\biggl{(}\frac{\partial}{\partial
t}\biggr{)}^{i}\exp{\biggl{(}\sum_{j=1}^{i}x_{j}\frac{t_{j}}{j!}\biggr{)}\bigg{|}_{t=0}}.$
(115)

This polynomial can also be expressed as an infinite sum of the partial
exponential Bell polynomial

$B_{i}(x_{1},\ldots,x_{i})=\sum_{j=1}^{i}{B_{i,j}(x_{1},\ldots,x_{i-j+1})},$
(116)

with

$B_{i,j}(x_{1},\ldots,x_{i-j+1})=\sum_{\\{k\\}}{\frac{i!}{k_{1}!\cdots
k_{i-j+1}!}\prod_{l=1}^{i-j+1}{\Bigl{(}\frac{x_{l}}{l!}\Bigr{)}^{k_{l}}}},$
(117)

where $\\{k\\}$ means that the sum is taken over all sequences $\\{k_{1}\sim
k_{i-j+1}\\}$ which satisfy

$\sum_{l=1}^{i-j+1}{k_{l}}=j,\qquad\sum_{l=1}^{i-j+1}{l\,k_{l}}=i.$ (118)

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