# Testing the microstructure of $d$-dimensional charged Gauss-Bonnet anti-de
Sitter black holes

Shao-Wen Wei<EMAIL_ADDRESS>corresponding author, Yu-Xiao Liu
<EMAIL_ADDRESS>Lanzhou Center for Theoretical Physics, Key Laboratory of
Theoretical Physics of Gansu Province, School of Physical Science and
Technology, Lanzhou University, Lanzhou 730000, People’s Republic of China,
Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China,
Joint Research Center for Physics, Lanzhou University and Qinghai Normal
University, Lanzhou 730000 and Xining 810000, China

###### Abstract

Understanding black hole microstructure via the thermodynamic geometry can
provide us with more deeper insight into black hole thermodynamics in modified
gravities. In this paper, we study the black hole phase transition and
Ruppeiner geometry for the $d$-dimensional charged Gauss-Bonnet anti-de Sitter
black holes. The results show that the small-large black hole phase transition
is universal in this gravity. By reducing the thermodynamic quantities with
the black hole charge, we clearly exhibit the phase diagrams in different
parameter spaces. Of particular interest is that the radius of the black hole
horizon can act as the order parameter to characterize the black hole phase
transition. We also disclose that different from the five-dimensional neutral
black holes, the charged ones allow the repulsive interaction among its
microstructure for small black hole of higher temperature. Another significant
difference between them is that the microscopic interaction changes during the
small-large black hole phase transition for the charged case, where the black
hole microstructure undergoes a sudden change. These results are helpful for
peeking into the microstructure of charged black holes in the Gauss-Bonnet
gravity.

Black holes, thermodynamics, phase transition, Ruppeiner geometry,
fluctuations

###### pacs:

04.70.Dy, 05.70.Ce, 04.50.Kd

##  I Introduction

Extended thermodynamics is one of the most active areas in black hole physics.
In particular, in the extended phase space, the cosmological constant is
interpreted as the pressure and its conjugate quantity is treated as the
thermodynamic volume Kastor . Although the black hole phase transition has
been observed in Refs. Chamblin ; Chamblin2 ; Cognola . The precise analogy
between the black hole and the van der Waals (VdW) fluid was completed in
Kubiznak . Reminiscent of the liquid-gas phase transition, the small-large
black hole phase transition was found to universally exist in the anti-de
Sitter (AdS) black holes. Besides, more interesting phase transition, such as
the reentrant phase transition, triple point, superfluid phase were observed
revealing that black hole has rich thermodynamic phenomena Kubiznak ;
Altamirano ; AltamiranoKubiznak ; Altamirano3 ; Dolan ; Liu ; Wei2 ; Frassino
; Cai ; XuZhao ; Hennigar ; Hennigar2 ; Tjoa2 ; Ruihong .

As we know, black holes have well defined macroscopic thermodynamic quantities
Hawking ; Bekensteina ; Bekensteinb , such as the temperature, entropy and so
on. Four thermodynamic laws of black holes also hold Bardeen . However, black
hole microstructure is a huge challenge. String theory and fuzzball model
showed some fruitful results, while plenty of fundamental problems are still
waiting for solved. How to test the microscopic properties of black hole by
using these macroscopic thermodynamic quantities is unknown.

Ruppeiner geometry developed in Ruppeiner  provides us with a natural approach
to peek into the microstructure of a thermodynamic system. It starts with the
Boltzmann entropy formula, and constructs the corresponding geometry in a
parameter space. Applying the similar treatment with the Riemannian geometry,
the scalar curvature can be produced. After the systematic study of a number
of fluid systems, some observations underlying the geometry were uncovered.
Here we summarize them as follows Ruppeiner : i) The line element of the
geometry measures the distance between two neighbouring fluctuation states,
and the less probable a fluctuation between two neighboring states, the longer
the distance between them; ii) The sign of the scalar curvature can be used to
test the interaction among the micromolecules of a thermodynamic system.
Positive and negative scalar curvatures are related to repulsive and
attractive interactions, respectively; iii) The scalar curvature of the
geometry corresponds to the correlation length near a critical point; iv)
Critical phenomena exists for such geometry.

Ruppeiner geometry has been introduced in the study of the black hole physics
CaiCho . However, as we shown above, it is closely related with the study of
the phase transition, in particular the VdW-like one. However, most early
studies focus on the divergent behaviors of the scalar curvature, as well as
that of the heat capacity.

In the extended phase space, the small-large black hole phase transition was
observed in many black hole backgrounds. At the critical point, the black
holes and the VdW fluid share the same critical exponents. It is extensively
known that phase transition is resulted by the coexistence and competition of
the microcosmic constituents of a thermodynamic system. So the microscopic
properties of the black hole can be exposed via the phase transition. More
importantly, this provides an excellent opportunity for the Ruppeiner
geometry.

Combining with the small-large black hole phase transition in a charged AdS
black hole background, the Ruppeiner geometry was constructed. Via the
corresponding scalar curvature, the black hole microstructure was first tested
Weiw . Further, we reconstructed the Ruppeiner geometry for the black hole
systems in a more natural approach Weiwa2 ; WeiWeiWei . First, we applied it
to the VdW fluid. Excluding the coexistence region where the equation of state
does not hold, we found that there is only the attractive interaction among
the fluid molecules. However, when applying it to the four-dimensional charged
AdS black hole, the vanishing heat capacity will spoil the geometry. To solve
this problem, we introduced a new normalized scalar curvature. It suggests
that repulsive interaction can exist for the small black hole with high
temperature, quite different from the VdW fluid. This uncovers the particular
microstructure of the black hole. Moreover, the critical phenomena of the
normalized scalar curvature were observed. Generalizing the study to higher
dimensions, we obtained the similar results. Subsequently, the approach has
been extended to other black hole backgrounds, and some interesting
microscopic properties of the black hole microstructure were disclosed
Dehyadegari ; Zangeneh:2016fhy ; Moumni ; Miao ; Du ; Xuz ; GhoshBhamidipati ;
Kumara2020 ; Yerra:2020oph ; Wu:2020fij ; Vaid ; Rizwan ; Mansoori ; Kumara ;
Mannw ; Dehyadegariw ; Wei2020d ; Wei2020a ; Zhourun ; YPhu ; Xuzzm .

On the other hand, the Gauss-Bonnet (GB) gravity is a widely concerned
modified gravity. The thermodynamics and phase transition of the black holes
were extensively investigated Liu ; Wei2 ; Cai . In Refs. Liu ; Wei2 , we
first observed the triple point in six-dimensional charged GB-AdS black holes.
While in other dimensions, only the small-large black hole phase transition
was found. More interestingly, for the five-dimensional neutral GB-AdS black
hole, the coexistence curve of the small and large black holes was
analytically obtained Mol . For the charged case, the analytic coexistence
curve is gained in the grand canonical ensemble Zhourun . These provide us
with a good opportunity to exactly study its thermodynamics.

For the black holes in the four-dimensional GB gravity, we found that there
are the small-large black hole phase transition both in the canonical and
grand canonical ensembles Wei2020d . After constructing the Ruppeiner
geometry, we found that its microscopic properties are similar to the charged
AdS black hole in Einstein gravity. For the small black hole with high
temperature, the repulsive interaction dominates among the microstructure.

However, for the five-dimensional neutral GB-AdS black hole, the result
behaves quite interesting Wei2020a . When excluding the coexistence regions,
only the attractive interaction is allowed. This suggests that the repulsive
interaction is not necessary for the black holes. Making use the analytical
coexistence curve, we observed that along the coexistence small and large
black hole curves, the normalized scalar curvature keeps the same values. As
we know, when a phase transition takes place, the microstructure gets a sudden
change. The value of the scalar curvature is generally believed to indicate
the interaction, and same values reflect the same interaction. So for the
five-dimensional neutral black hole, a significant property reflected from the
Ruppeiner geometry is that its microscopic interaction keeps the same while
its microstructure suddenly changes. This uncovers the unique nature of the GB
gravity. This particular property was also observed in the grand canonical
ensemble Zhourun .

Since the charged case in the GB gravity has not been studied in $d\geq$5 in
the canonical ensemble, in this paper, we aim to study the thermodynamics and
microstructure for them and to see whether an interesting different
microstructure can be revealed. The present paper is organized as follows. In
Sec. II, we briefly review the thermodynamics and Ruppeiner geometry for the
charged GB-AdS black holes. In Sec. III, we consider the black hole in five
dimensions. The phase diagram and the scalar curvature are given. The critical
phenomena are also numerically investigated. Then in Sec. IV, we generalize
the study to the black holes in $d\geq$6\. Finally, we summarize and discuss
our results.

##  II Thermodynamics and microstructure of charged Gauss-Bonnet AdS black
holes

In this section, we shall introduce the thermodynamics and Ruppeiner geometry
for the $d$-dimensional charged GB-AdS black holes.

###  II.1 Thermodynamics

The line element for the black hole is Boulware ; Cai2 ; Wiltshire ; Cvetic2

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2}_{d-2},$ (1)
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{64\pi\alpha
M}{(d-2)\omega_{d-2}r^{d-1}}-\frac{8\alpha
Q^{2}}{(d-2)(d-3)r^{2d-4}}-\frac{64\pi\alpha P}{(d-1)(d-2)}}\right),$ (2)

where $M$ and $Q$ are the mass and charge of the black hole, respectively. The
parameter $\alpha$ is related with the GB coupling
$\alpha=(d-3)(d-4)\alpha_{\text{GB}}$ with dimension [length]2. The volume of
the unit $(d-2)$-sphere is $\omega_{d-2}=2\pi^{(d-1)/2}/\Gamma((d-1)/2)$. Here
we consider the thermodynamic quantity in the extended phase space, where the
cosmological constant $\Lambda$ has been interpreted as the pressure
$P=-\frac{\Lambda}{8\pi}$. This solution is obtained from the following action

$\displaystyle S=\frac{1}{16\pi}\int
d^{d}x\sqrt{-g}\left(R-2\Lambda+\alpha_{\text{GB}}\mathcal{L}_{\text{GB}}-4\pi
F_{\mu\nu}F^{\mu\nu}\right),$ (3)

where the GB Lagrangian
$\mathcal{L}_{\text{GB}}=\mathcal{R}_{\mu\nu\gamma\delta}\mathcal{R}^{\mu\nu\gamma\delta}-4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}+\mathcal{R}^{2}$.

The radius of the event horizon can be obtained by solving
$f(r_{\text{h}})=0$, which depends on the black hole mass, charge and
pressure. Here we can also express the black hole mass as

$\displaystyle
M=\frac{(d-2)\omega_{d-2}r_{\text{h}}^{d-3}}{16\pi}\left(1+\frac{\alpha}{r_{\text{h}}^{2}}+\frac{16\pi
Pr_{\text{h}}^{2}}{(d-1)(d-2)}\right)+\frac{\omega_{d-2}Q^{2}}{8\pi(d-3)r_{\text{h}}^{d-3}}.$
(4)

Other thermodynamic quantities, such as the temperature $T$, entropy $S$,
electric potential $\Phi$, and conjugate quantity $\mathcal{A}$ to $\alpha$
are given by Cai

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{-2Q^{2}r_{\text{h}}^{7-2d}+2(d-3)(d-2)r_{\text{h}}+32\pi
Pr_{\text{h}}^{3}}{4\pi(d-2)\left(2\alpha+r_{\text{h}}^{2}\right)}-\frac{5\alpha-\alpha
d}{8\pi\alpha r_{\text{h}}+4\pi r_{\text{h}}^{3}},$ (5) $\displaystyle S$
$\displaystyle=$
$\displaystyle\frac{\omega_{d-2}r_{\text{h}}^{d-2}}{4}\left(1+\frac{2(d-2)\alpha}{(d-4)r_{\text{h}}^{2}}\right),$
(6) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\frac{\omega_{d-2}Q}{4\pi(d-3)r_{\text{h}}^{d-3}},$ (7)
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{d-2}r_{\text{h}}^{-d-5}\left(r_{\text{h}}^{2d}\left(-2\alpha(d-2)+(d-2)^{2}r_{\text{h}}^{2}+32\pi
Pr_{\text{h}}^{4}\right)-4Q^{2}r_{\text{h}}^{8}\right)}{16\pi(d-4)\left(2\alpha+r_{\text{h}}^{2}\right)}.$
(8)

The specific volume and thermodynamic volume are given by

$\displaystyle v$ $\displaystyle=$ $\displaystyle\frac{4r_{\text{h}}}{d-2},$
(9) $\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{\omega_{d-2}r_{\text{h}}^{d-1}}{d-1}.$ (10)

It is clear that $V\sim v^{d-1}$, so they are not linear, which is quite
different from the VdW fluid. For the black hole thermodynamics, these two
volumes have different applications, see the related discussion in Ref. Wei3 .
One can easily confirm that the first law and the Smarr formula hold

$\displaystyle dM$ $\displaystyle=$ $\displaystyle TdS+\Phi
dQ+\mathcal{A}d\alpha+VdP,$ (11) $\displaystyle(d-3)M$ $\displaystyle=$
$\displaystyle(d-2)TS+(d-3)Q\Phi+2\mathcal{A}\alpha-2PV.$ (12)

From (11), we find that the pressure-volume term is $VdP$ rather than $-PdV$.
So, as suggested in Ref. Kastor , the black hole mass $M$ here should be
treated as the enthalpy $H$ of the system.

Solving Eq. (5), we can obtain the equation of state

$\displaystyle P=\frac{T}{v}-\frac{d-3}{(d-2)\pi
v^{2}}+\frac{32T\alpha}{(d-2)^{2}v^{3}}-\frac{16\alpha(d-5)}{(d-2)^{3}\pi
v^{4}}+\frac{2^{4d-11}Q^{2}}{\pi(d-2)^{2d-4}v^{2d-4}}.$ (13)

Obviously, this equation closely depends on the GB coupling $\alpha$ and
charge $Q$. According to the classification of the black hole quantities Wei6
, this black hole system is a double characteristic parameter system. However,
we can reduce these thermodynamic quantities with the charge $Q$. This shall
help us to simplify the study. Via the dimensional analysis, we adopt the
following reduced forms

$\displaystyle\tilde{T}=TQ^{\frac{1}{d-3}},\quad\tilde{P}=PQ^{\frac{2}{d-3}},\quad\tilde{r}_{\text{h}}=r_{\text{h}}Q^{-\frac{1}{d-3}},\quad\tilde{v}=vQ^{-\frac{1}{d-3}},\quad\tilde{V}=VQ^{-\frac{d-1}{d-3}}.$
(14)

Then the reduced equation of state reads

$\displaystyle\tilde{P}=\frac{\tilde{T}}{\tilde{v}}-\frac{d-3}{(d-2)\pi\tilde{v}^{2}}+\frac{32\tilde{T}\tilde{\alpha}}{(d-2)^{2}\tilde{v}^{3}}-\frac{16\tilde{\alpha}(d-5)}{(d-2)^{3}\pi\tilde{v}^{4}}+\frac{2^{4d-11}}{\pi(d-2)^{2d-4}\tilde{v}^{2d-4}}.$
(15)

Now we find that the equation of state only depends on the reduced GB coupling
$\tilde{\alpha}=\alpha/Q^{2/(d-3)}$. Moreover, we can express the reduced
pressure in terms of the thermodynamic volume $\tilde{V}$. Another important
thermodynamic quantity is the Gibbs free energy, which is

$\displaystyle\tilde{G}=\frac{H-TS}{Q}$ $\displaystyle=$
$\displaystyle\frac{\omega_{d-2}}{16\pi(d-4)(d-3)(d-2)(d-1)\left(2\alpha+\tilde{r}_{\text{h}}^{2}\right)\tilde{r}_{\text{h}}^{d+5}}\times\bigg{(}(d-3)\tilde{r}_{\text{h}}^{2d}\bigg{(}2\tilde{\alpha}^{2}(d-2)^{2}(d-1)$
(16) $\displaystyle+$
$\displaystyle\tilde{\alpha}(d-2)\tilde{r}_{\text{h}}^{2}\left((d-9)d-96\pi\tilde{P}\tilde{r}_{\text{h}}^{2}+8\right)+(d-4)\tilde{r}_{\text{h}}^{4}\left((d-3)d-16\pi\tilde{P}\tilde{r}_{\text{h}}^{2}+2\right)\bigg{)}$
$\displaystyle+$ $\displaystyle
2(d-1)\tilde{r}_{\text{h}}^{8}\left(2\tilde{\alpha}(d-2)(2d-7)+(d-4)(2d-5)\tilde{r}_{\text{h}}^{2}\right)\bigg{)}.$

The local thermodynamic stability is measured by the heat capacity. Positive
or negative one indicates the system is local stable or unstable. The heat
capacity can be calculated by

$\displaystyle C_{Q}=T\left(\frac{\partial S}{\partial
T}\right)_{Q,\alpha,P}.$ (17)

Since it is in a complex form, we shall not show it here. In the following
section, we will investigate the phase structure for different dimension
number $d$.

###  II.2 Ruppeiner geometry and microstructure

In this subsection, we would like to give a brief introduction for the
Ruppeiner geometry, and to show how black hole microstructure is uncovered via
the geometry.

Here we consider that the system S we concerned is surrounded by its huge
environment E. Then the total entropy of the complete system can be written as

$S(x^{0},x^{1})=S_{\rm S}(x^{0},x^{1})+S_{\rm E}(x^{0},x^{1}),$ (18)

where we assume the system has two independent thermodynamic extensive
variables $x^{0}$ and $x^{1}$. Moreover, $S_{\rm S}\ll S_{\rm E}\sim S$ is
required. Near the local maximum of the entropy at $x^{\mu}=x^{\mu}_{0}$, we
expand the total entropy as

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{0}+\left.\frac{\partial
S_{\rm S}}{\partial x^{\mu}}\right|_{x^{\mu}_{0}}\Delta x^{\mu}_{\rm
S}+\left.\frac{\partial S_{\rm E}}{\partial
x^{\mu}}\right|_{x^{\mu}_{0}}\Delta x^{\mu}_{\rm E}$ (19)
$\displaystyle+\left.\frac{1}{2}\frac{\partial^{2}S_{\rm S}}{\partial
x^{\mu}\partial x^{\nu}}\right|_{x^{\mu}_{0}}\Delta x^{\mu}_{\rm S}\Delta
x^{\nu}_{\rm S}+\left.\frac{1}{2}\frac{\partial^{2}S_{\rm E}}{\partial
x^{\mu}\partial x^{\nu}}\right|_{x^{\mu}_{0}}\Delta x^{\mu}_{\rm E}\Delta
x^{\nu}_{\rm E}+\cdots,\quad(\mu,\nu=0,1),$

where we denote the local maximum of the entropy $S(x^{\mu}_{0})$ with
$S_{0}$. Since the fluctuating parameters are additive, i.e.,
$x_{S}^{\mu}+x_{E}^{\mu}=x_{total}^{\mu}=constant$, one can easily get
$\left.\frac{\partial S_{\rm S}}{\partial x^{\mu}}\right|_{0}\Delta
x^{\mu}_{\rm S}+\left.\frac{\partial S_{\rm E}}{\partial
x^{\mu}}\right|_{0}\Delta x^{\mu}_{\rm E}=0$. Thus we have $\Delta S=S-S_{0}$,

$\Delta S=\left.\frac{1}{2}\frac{\partial^{2}S_{\rm S}}{\partial
x^{\mu}\partial x^{\nu}}\right|_{0}\Delta x^{\mu}_{\rm S}\Delta x^{\nu}_{\rm
S}+\left.\frac{1}{2}\frac{\partial^{2}S_{\rm E}}{\partial x^{\mu}\partial
x^{\nu}}\right|_{0}\Delta x^{\mu}_{\rm E}\Delta x^{\nu}_{\rm E}+\cdots\;.$
(20)

The second term is much smaller than the first term, so we ignore it. Finally,
the probability of finding the system in the internals $x_{0}\sim(x_{0}+\Delta
x_{0})$ and $x_{1}\sim(x_{1}+\Delta x_{1})$ can be expressed in the following
form

$\displaystyle P(x^{0},x^{1})\propto e^{-\frac{1}{2}\Delta l^{2}},$ (21)

where

$\displaystyle\Delta l^{2}$ $\displaystyle=$ $\displaystyle-\frac{1}{k_{\rm
B}}g_{\mu\nu}\Delta x^{\mu}\Delta x^{\nu},$ (22) $\displaystyle g_{\mu\nu}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial^{2}S_{\rm S}}{\partial
x^{\mu}\partial x^{\nu}}\right|_{0}.$ (23)

According to the thermodynamic information geometry, $\Delta l^{2}$ measures
the distance between two neighboring fluctuation states. The less probable a
fluctuation between two neighboring states, the longer the distance between
them. Following the Riemannian geometry, we can construct the corresponding
scalar curvature for this Ruppeiner geometry. The empirical observation
suggests that positive (negative) scalar curvature indicates the repulsive
(attractive) interaction among the small molecules of the system. Further
study also uncovers that the scalar curvature is related to the correlation
length for the thermodynamic system near the critical point.

Taking the internal energy $U$ and thermodynamic volume $V$ as the fluctuation
coordinates, the line element can be reexpressed via the first law of the
black hole thermodynamics as

$dl^{2}=-\frac{1}{T}\left(\frac{\partial^{2}F}{\partial
T^{2}}\right)_{V}dT^{2}+\frac{1}{T}\left(\frac{\partial^{2}F}{\partial
V^{2}}\right)_{T}dV^{2},$ (24)

where the free energy $F=U-TS$ and it satisfies the differential law $dF=-SdT-
PdV$. By making use of the heat capacity at constant volume
$C_{V}=T(\partial_{T}S)_{V}$, the line element reads

$dl^{2}=\frac{C_{V}}{T^{2}}dT^{2}-\frac{(\partial_{V}P)_{T}}{T}dV^{2}.$ (25)

The scalar curvature for this geometry is WeiWeiWei

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2C_{V}^{2}(\partial_{V}P)^{2}}\bigg{\\{}T(\partial_{V}P)\bigg{[}(\partial_{T}C_{V})(\partial_{V}P-T\partial_{T,V}P)+(\partial_{V}C_{V})^{2}\bigg{]}$
$\displaystyle+$ $\displaystyle
C_{V}\bigg{[}(\partial_{V}P)^{2}+T\left((\partial_{V}C_{V})(\partial_{V,V}P)-T(\partial_{T,V}P)^{2}\right)+2T(\partial_{V}P)(T(\partial_{T,T,V}P)-(\partial_{V,V}C_{V}))\bigg{]}\bigg{\\}}.$

Similar to the charged AdS black holes, here we also have $C_{V}$=0 for the
charged GB-AdS black holes. This will lead to the divergence of $R$. In order
to uncover the specific properties hiding behind it, we have defined a new
normalized scalar curvature, which is

$R_{\rm
N}=R*C_{V}=\frac{1}{2}-\frac{T^{2}(\partial_{V,T}P)^{2}-2T^{2}(\partial_{V}P)(\partial_{V,T,T}P)}{2(\partial_{V}P)^{2}}.$
(27)

Adopting this normalized scalar curvature, the microstructure has been
explored for several different black holes.

##  III Phase diagram and microstructure in five dimensions

In this section, we shall focus on the five dimensions with $d$=5 and
$\omega_{3}=2\pi^{2}$. In Refs. Wei2 ; Cai , it was shown that there exists
the small-large black hole phase transition. Here we shall examine its phase
diagrams in detailed and construct the correspond Ruppeiner geometry.

###  III.1 Phase diagrams and phase transitions

For $d$=5, the reduced equation of state, the Gibbs free energy, and the heat
capacity are given by

$\displaystyle\tilde{P}$ $\displaystyle=$
$\displaystyle\frac{3\tilde{T}}{4\tilde{r}_{\text{h}}}-\frac{3}{8\pi\tilde{r}_{\text{h}}^{2}}+\frac{3\tilde{\alpha}\tilde{T}}{2\tilde{r}_{\text{h}}^{3}}+\frac{1}{8\pi\tilde{r}_{\text{h}}^{6}},$
(28) $\displaystyle\tilde{G}$ $\displaystyle=$
$\displaystyle-\frac{\pi\left(2\pi\tilde{T}\tilde{r}_{\text{h}}^{5}-3\tilde{r}_{\text{h}}^{4}+36\pi\tilde{\alpha}\tilde{T}\tilde{r}_{\text{h}}^{3}-6\tilde{\alpha}\tilde{r}_{\text{h}}^{2}-3\right)}{16\tilde{r}_{\text{h}}^{2}},$
(29) $\displaystyle C_{Q}/Q^{3/2}$ $\displaystyle=$
$\displaystyle\frac{3\pi^{2}\tilde{r}_{\text{h}}\left(8\pi
P\tilde{r}_{\text{h}}^{6}+3\tilde{r}_{\text{h}}^{4}-1\right)\left(2\tilde{\alpha}+\tilde{r}_{\text{h}}^{2}\right)^{2}}{2\left(6\tilde{\alpha}+3\tilde{r}_{\text{h}}^{6}(16\pi\tilde{\alpha}\tilde{P}-1)+8\pi\tilde{P}\tilde{r}_{\text{h}}^{8}+6\tilde{\alpha}\tilde{r}_{\text{h}}^{4}+5\tilde{r}_{\text{h}}^{2}\right)}$
(30)

The equation of state (28) exhibits a small-large black hole phase transition,
reminiscent of the liquid-gas phase transition of the VdW fluid. For the phase
transition, there is a characteristic critical point indicating a second-order
phase transition, which can be determined by

$\displaystyle\left(\frac{\partial P}{\partial
V}\right)_{T,Q,\alpha}=0,\quad\left(\frac{\partial^{2}P}{\partial
V^{2}}\right)_{T,Q,\alpha}=0.$ (31)

Since $V=\frac{\pi^{2}}{2}r_{\text{h}}^{4}$ for $d$=5, the conditions can be
expressed as

$\displaystyle\left(\frac{\partial\tilde{P}}{\partial\tilde{r}_{\text{h}}}\right)_{\tilde{T},\tilde{\alpha}}=0,\quad\left(\frac{\partial^{2}\tilde{P}}{\partial\tilde{r}_{\text{h}}^{2}}\right)_{\tilde{T},\tilde{\alpha}}=0.$
(32)

Solving the equation, we have

$\displaystyle
4\tilde{r}_{\text{h}}^{6}-24\tilde{\alpha}\tilde{r}_{\text{h}}^{4}-5\tilde{r}_{\text{h}}^{2}-18\tilde{\alpha}=0.$
(33)

Then the critical value of the radius of the horizon can be obtained

$\displaystyle\tilde{r}_{\text{hc}}$ $\displaystyle=$
$\displaystyle\sqrt{2\alpha+2\sqrt{\frac{5+12\tilde{\alpha}^{2}}{3}}\cos\Theta},$
(34) $\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\frac{1}{3}\arccos\left(\frac{6\sqrt{3}\tilde{\alpha}(7+4\tilde{\alpha}^{2})}{(5+12\tilde{\alpha}^{2})^{\frac{3}{2}}}\right).$
(35)

The corresponding critical pressure and temperature are

$\displaystyle\tilde{P}_{\text{c}}$ $\displaystyle=$
$\displaystyle-\frac{6\tilde{\alpha}+6\tilde{\alpha}\tilde{r}_{\text{hc}}^{4}-3\tilde{r}_{\text{hc}}^{6}+5\tilde{r}_{\text{hc}}^{2}}{48\pi\tilde{\alpha}\tilde{r}_{\text{hc}}^{6}+8\pi\tilde{r}_{\text{hc}}^{8}},$
(36) $\displaystyle\tilde{T}_{\text{c}}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{\text{hc}}^{4}-1}{\pi\tilde{r}_{\text{hc}}^{3}\left(6\tilde{\alpha}+\tilde{r}_{\text{hc}}^{2}\right)}.$
(37)

We plot the critical point in the $\tilde{P}$-$\tilde{T}$ diagram in Fig. 1
for varying $\tilde{\alpha}$. When $\tilde{\alpha}$=0, the critical point
curve starts at ($\frac{4}{5\sqrt[4]{5}\pi}$, $\frac{1}{4\sqrt{5}\pi}$). Then
with the increase of $\tilde{\alpha}$, the curve extends to low temperature
and pressure, and ends at (0, 0) when $\tilde{\alpha}\rightarrow\infty$. Each
point on this curve is a phase transition point of second-order.

Figure 1: Critical points in the $\tilde{P}$-$\tilde{T}$ diagram.
$\tilde{\alpha}$ increases from the right to left. The left and right dots are
for $\tilde{\alpha}$=0 and $\infty$.

Figure 2: Heat capacity as a function of the temperature at fixed critical
pressure. The divergent behaviors occur at the critical temperature. (a)
$\tilde{\alpha}$=0.4. (b) $\tilde{\alpha}$=1.4. Obviously, the $\lambda$ phase
transition behavior is observed.

For $\tilde{\alpha}$=0.4 and 1.4, we plot the heat capacity in Fig. 2 when the
pressure takes its critical values. Obviously, we observe the $\lambda$
behaviors for these two cases. Such pattern of the heat capacity is
reminiscent of that of 4He. For black hole systems, this behavior was first
observed in Ref. Tjoa2 , where a new superfluid was uncovered. Here, we
suggest that such $\lambda$ behavior of the heat capacity implies a second-
order phase transition rather a new black hole phase unless the conditions
given in Ref. Tjoa2  are satisfied.

Employing these critical thermodynamic quantities, we can construct a
universal quantity

$\displaystyle\rho=\frac{\tilde{P}_{\text{c}}\tilde{r}_{\text{hc}}}{\tilde{T}_{\text{c}}}=\frac{6\tilde{\alpha}-3\tilde{r}_{\text{hc}}^{6}+6\tilde{\alpha}\tilde{r}_{\text{hc}}^{4}+5\tilde{r}_{\text{hc}}^{2}}{8\tilde{r}_{\text{hc}}^{2}(1-\tilde{r}_{\text{hc}}^{4})}.$
(38)

For $\alpha$=0, we have $\rho$=5/16.

Next, we plot the isothermal curves and the Gibbs free energies for further
understanding the black hole phase transition. The isothermal curves are shown
in Fig. 3 in the $\tilde{P}$-$\tilde{V}$ diagram for $\tilde{\alpha}$=0.4 and
1.4, respectively. From these figures, one can observe nonmonotonic behavior
for low temperature, which indicates the existence of the small-large phase
transition. For fixed temperature, these black hole branches of the positive
slope are nonphysical. They should be removed by other mechanisms, one of
which is the Maxwell equal area law. One can draw a horizontal line to
construct two equal areas on one isothermal curve. Then the pressure and
temperature corresponding to the horizontal line denote the values of the
phase transition point. In our reduced treatment, it is equivalent to fix the
charge, so the equal area law is also applicable in the
$\tilde{P}$-$\tilde{V}$ diagram on determining the phase transition point.

Figure 3: The isothermal curves for the black hole in the
$\tilde{P}$-$\tilde{V}$ diagram for (a) $\tilde{\alpha}$=0.4.
$\tilde{T}$=0.091, 0.092, 0.093261 ($\tilde{T}_{\text{c}}$), and 0.094 from
bottom to top. (b) $\tilde{\alpha}$=1.4. $\tilde{T}$=0.053, 0.0536, 0.054209
($\tilde{T}_{\text{c}}$), and 0.055 from bottom to top.

Figure 4: Behaviors of the Gibbs free energy for fixed temperature. a)
$\tilde{\alpha}$=0.4. $\tilde{T}$=0.091, 0.092, 0.093261
($\tilde{T}_{\text{c}}$), and 0.094 from left to right. (b)
$\tilde{\alpha}$=1.4. $\tilde{T}$=$\tilde{T}$=0.053, 0.0536, 0.054209
($\tilde{T}_{\text{c}}$), and 0.055 from left to right.

Another alternative on determining the phase transition is the Gibbs free
energy. We show its behavior in Fig. 4. When the temperature is lower than the
critical value, there is the characteristic swallow tail behavior. Combining
the thermodynamic choice that a system always prefers the lowest free energy,
the phase transition exactly occurs at the intersection point of the Gibbs
free energy. Therefore, one can also use such property to numerically solve
the phase transition point.

Actually, these two approaches coincide with each other when the first law of
the black hole thermodynamics holds. Here we display the phase structures in
the $\tilde{P}$-$\tilde{T}$ and $\tilde{T}$-$\tilde{r}_{\text{h}}$ diagrams in
Figs. 5 and 6, respectively. In the $\tilde{P}$-$\tilde{T}$ diagram shown in
Fig. 5, the red solid curves are for the coexistence phase of small and large
black holes. Above the curve is the small black hole region and below it is
the large black hole region. Black dots denote the critical points, where the
second-order phase transition occurs. Moreover, we also observe that the
critical point is shifted toward low temperature and pressure for big
$\tilde{\alpha}$. In Fig. 6, the phase diagram is described in the
$\tilde{T}$-$\tilde{r}_{\text{h}}$ diagram. Obviously, the coexistence curve
given in Fig. 5 becomes the coexistence region, see below the red curve. The
small and large black hole regions, respectively, locate at the left and
right. For different $\tilde{\alpha}$, the pattern is similar, while with
different peaks indicating different critical points.

Figure 5: Phase structures in $\tilde{P}$-$\tilde{T}$ diagram. (a)
$\tilde{\alpha}$=0.4. (b) $\tilde{\alpha}$=1.4. SBH and LBH are for the small
and large black holes. Black dots denote the critical points.

Figure 6: Phase structures in the $\tilde{T}$-$\tilde{r}_{\text{h}}$ diagram.
(a) $\tilde{\alpha}$=0.4. (b) $\tilde{\alpha}$=1.4.

As shown in Fig. 6, the radius of the black hole horizon has a sudden change
for a fixing phase transition temperature, which indicates that black hole
microstructure gets a significant change. Now we define two new radii shifted
by the critical value of the horizon radius at the phase transition

$\displaystyle\tilde{r}_{\text{h1}}=\tilde{r}_{\text{hl}}-\tilde{r}_{\text{hc}},\quad\tilde{r}_{\text{h2}}=\tilde{r}_{\text{hc}}-\tilde{r}_{\text{hs}},$
(39)

where $\tilde{r}_{\text{hs}}$ and $\tilde{r}_{\text{hl}}$ denote the horizon
radius of the coexistence small and large black holes. We describe
$\tilde{r}_{\text{h1}}$ and $\tilde{r}_{\text{h2}}$ in Fig. 7. From Fig. 7, we
find that $\tilde{r}_{\text{h1}}$ has large values near $\tilde{T}$=0. Then it
decreases with $\tilde{T}$. At the critical temperature,
$\tilde{r}_{\text{h1}}$ vanishes for $\tilde{\alpha}$=0.4 and 1.4. Similar
phenomena can also be observed for $\tilde{r}_{\text{h2}}$, see in Fig. 7. The
only difference is that near $\tilde{T}$=0, $\tilde{r}_{\text{h2}}$ is finite.
For example, $\tilde{r}_{\text{h2}}$=1.25 and 2.28 for $\tilde{\alpha}$=0.4
and 1.4, respectively. The reason is that $\tilde{r}_{\text{hl}}$ has no
boundary, while $\tilde{r}_{\text{hs}}$ cannot be smaller than zero.

Figure 7: Behaviors of $\tilde{r}_{\text{h1}}$ (a) and $\tilde{r}_{\text{h2}}$
(b). Red solid curves are for $\tilde{\alpha}$=0.4 and blue dashed curves are
for $\tilde{\alpha}$=1.4.

As shown above $\tilde{r}_{\text{h1}}$ and $\tilde{r}_{\text{h2}}$ vanish at
the critical temperature, now, we aim to calculate its critical exponent.
After the numerical calculation near $\tilde{T}$=0, we fit the data with the
following form

$\displaystyle\tilde{r}_{\text{h1,2}}=a(\tilde{T}_{\text{c}}-\tilde{T})^{b},$
(40)

where $a$ and $b$ are the fitting coefficients. We show the critical points
and fitting coefficients in Table 1. Focusing on the coefficient $b$, we find
that all them are near 1/2 with the relative deviation within 1.6%. Actually
this value can also be obtained via expanding the equation of state of the
system near the critical point. So we can conclude that
$\tilde{r}_{\text{h1}}$ and $\tilde{r}_{\text{h2}}$ have a critical exponent
1/2. Hence $\tilde{r}_{h1,2}$, or the horizon radius, can serve as order
parameters to characterize the small-large black hole phase transition. This
also provides a fundamental interpretation of the horizon radius on
investigating the dynamic process of the phase transitions.

$\alpha$ | $\tilde{P}_{\text{c}}$ | $\tilde{T}_{\text{c}}$ | $\tilde{r}_{\text{hc}}$ | $a$ | $b$
---|---|---|---|---|---
0.4 | 0.012748 | 0.093261 | 2.016305 | 12.8281/10.0179 | 0.5095/0.4903
1.4 | 0.004560 | 0.054209 | 3.039192 | 32.6572/26.0413 | 0.5078/0.4913
Table 1: Values of the critical points and fitting coefficients for
$\tilde{\alpha}$=0.4 and $\tilde{\alpha}$=1.4. …/… means the data are for
$\tilde{r}_{\text{h1}}$/$\tilde{r}_{\text{h2}}$.

###  III.2 Ruppeiner geometry and microstructure

As observed in Refs. Wei2020a ; Zhourun , the neutral GB-AdS black hole has an
intriguing property that its interaction among the microscopic constituents
maintains the same while its microstructure behaves different before and after
the small-large black hole phase transition. In this subsection, we shall
study the microstructure of the black hole in five dimensions and see whether
this property holds for the charged case in the canonical ensemble.

Making use of the equation of state (28), we obtain the corresponding
normalized scalar curvature of the Ruppeiner geometry via (27). When
expressing in terms of $\tilde{T}$ and $\tilde{r}_{\text{h}}$, we have

$\displaystyle
R_{\text{N}}=-\frac{\left(4\tilde{r}_{\text{h}}^{4}-1\right)\left(12\pi\tilde{\alpha}\tilde{T}\tilde{r}_{\text{h}}^{3}+2\pi\tilde{T}\tilde{r}_{\text{h}}^{5}-\tilde{r}_{\text{h}}^{4}+1\right)}{2\left(6\pi\tilde{\alpha}\tilde{T}\tilde{r}_{\text{h}}^{3}+\pi\tilde{T}\tilde{r}_{\text{h}}^{5}-\tilde{r}_{\text{h}}^{4}+1\right)^{2}}.$
(41)

Obviously, the charge is absent by using the reduced thermodynamic quantities.
We plot $R_{\text{N}}$ for different $\tilde{T}$ and $\tilde{r}_{\text{h}}$
for $\tilde{\alpha}=0.4$ in Fig. 8. Other values of $\tilde{\alpha}$ share the
similar pattern. For low temperature, $R_{\text{N}}$ has two negative
divergences at two $\tilde{r}_{\text{h}}$. These two divergences merger at the
critical point, and disappear beyond the critical temperature.

In order to show the details, we display $R_{\text{N}}$ for
$\tilde{T}$=0.4$\tilde{T}_{\text{c}}$, 0.8$\tilde{T}_{\text{c}}$,
$\tilde{T}_{\text{c}}$, and 1.2 $\tilde{T}_{\text{c}}$ in Fig. 9. It is clear
that there are two negative divergent points for
$\tilde{T}$=0.4$\tilde{T}_{\text{c}}$ and 0.8$\tilde{T}_{\text{c}}$, and one
for $\tilde{T}=\tilde{T}_{\text{c}}$. While when
$\tilde{T}=1.2\tilde{T}_{\text{c}}$, there is only a well with finite depth.
More importantly, for $\tilde{T}$=0.4$\tilde{T}_{\text{c}}$ shown in Fig. 9,
we find that there are two parameter regions, where $R_{\text{N}}$ is
positive. One of them is at small $\tilde{r}_{\text{h}}$. For different
temperatures, we always observe positive $R_{\text{N}}$. Another region is
near $\tilde{r}_{\text{h}}$=2. However, it only appears for
$\tilde{T}$=0.4$\tilde{T}_{\text{c}}$.

Figure 8: The normalized scalar curvature $R_{\text{N}}$ as a function of
$\tilde{T}$ and $\tilde{r}_{\text{h}}$ for $\tilde{\alpha}=0.4$.

Figure 9: Behaviors of $R_{\text{N}}$ for different fixing temperature with
$\tilde{\alpha}$=0.4. (a) $\tilde{T}$=0.4$\tilde{T}_{\text{c}}$. (b)
$\tilde{T}$=0.8$\tilde{T}_{\text{c}}$. (c) $\tilde{T}$=$\tilde{T}_{\text{c}}$.
(d) $\tilde{T}$=1.2$\tilde{T}_{\text{c}}$.

Now, we aim to uncover the divergent point and zero point of $R_{\text{N}}$.
Solving (41), we obtain the divergent temperature $\tilde{T}_{\text{sp}}$ of
$R_{\text{N}}$, which gives

$\displaystyle\tilde{T}_{\text{sp}}=\frac{\tilde{r}_{\text{h}}^{4}-1}{\pi\tilde{r}_{\text{h}}^{3}(\tilde{r}_{\text{h}}^{2}+6\tilde{\alpha})}.$
(42)

Actually, the curve described by this temperature separates the metastable and
unstable black hole branches, so we can call it the spinodal curve. The
temperature corresponding to zero point is

$\displaystyle\tilde{T}_{0}=\frac{1}{2}\tilde{T}_{\text{sp}}.$ (43)

Besides, at

$\displaystyle\tilde{r}_{\text{h}}=1,$ (44)

$R_{\text{N}}$ will also vanish. We name the zero point curve of
$R_{\text{N}}$ as the sign-changing point, across which $R_{\text{N}}$ changes
its sign.

Figure 10: The coexistence curves (red solid lines), spinodal curves (blue
dashed lines), and the sign-changing curves of $R_{\text{N}}$ (black dot-
dashed lines). The normalized scalar curvature $R_{\text{N}}$ diverges at the
spinodal curves. Both the spinodal and sign-changing curves start at
$\tilde{r}_{\text{h}}$=$1/\sqrt{2}$. In the shadow regions I, II, and III,
$R_{\text{N}}$ is positive, otherwise, it is negative. (a)
$\tilde{\alpha}$=0.4. (b) $\tilde{\alpha}$=1.4.

We describe these characteristic curves in the phase diagram in Fig. 10 for
$\tilde{\alpha}$=0.4 and 1.4, respectively. The red solid curves are the
coexistence curves of small and large black holes, below which is the
coexistence regions, and the equation of state (28) may not hold. So we need
to exclude such regions when we consider the normalized scalar curvature
$R_{\text{N}}$, where we have used the equation of state in the calculation.
The spinodal curves are marked in blue dashed lines. They only meet the
coexistence curves at the critical points. The black dot-dashed lines are for
the sign-changing curves, which separate the phase diagram into two regions,
one with positive $R_{\text{N}}$ and another with negative $R_{\text{N}}$
shown in shadow regions.

For each $\tilde{\alpha}$, we find that there are three regions I, II, and III
that have positive $R_{\text{N}}$, indicating that the repulsive interaction
dominates among the black hole microstructure. While in other regions, the
attractive interaction dominates. Since regions I and III are in the
coexistence regions, we exclude them in our consideration. However, in region
II, the small black holes with higher temperature always have positive
$R_{\text{N}}$. So the repulsive interaction can be observed for the charged
GB-AdS black holes. This behavior is quite similar to the charged AdS black
hole without the GB coupling WeiWeiWei , while different from the neutral GB-
AdS black holes Wei2020a ; Zhourun .

Figure 11: $R_{\text{N}}$ along the coexistence small (top blue curves) and
large (bottom red curves) black holes. Black dots denote the zero points of
$R_{\text{N}}$, which are at $\tilde{T}$=0.062312 and 0.028235, respectively.
(a) $\tilde{\alpha}$=0.4. (b) $\tilde{\alpha}$=1.4.

We also show the behaviors of $R_{\text{N}}$ along the coexistence small and
large black hole curves for $\tilde{\alpha}$=0.4 and 1.4 in Fig 11. With the
increase of the temperature, both the curves decrease, and go to negative
infinity at the critical temperature. In particular, $R_{\text{N}}$ is
positive at low temperature, $T<$0.062312 for $\tilde{\alpha}$=0.4 and
$T<$0.028235 for $\tilde{\alpha}$=1.4 when it goes along the coexistence small
black holes indicating that the repulsive interaction exists.

Figure 12: $\ln|R_{\text{N}}|$ vs. $\ln(\tilde{T}_{\text{c}}-\tilde{T})$ when
the temperature is very near its critical values. Bottom blue and top red
curves are for the coexistence small and large black holes. (a)
$\tilde{\alpha}$=0.4. (b) $\tilde{\alpha}$=1.4.

At the critical temperature, all these curves go to negative infinity. Here we
plot $\ln|R_{\text{N}}|$ as a function of
$\ln(\tilde{T}_{\text{c}}-\tilde{T})$ in Fig. 12 near the critical
temperature. An intuitive impression is that it has a linear relation. We fit
the data in the following form

$\displaystyle\ln|R_{\text{N}}|=\beta\ln(\tilde{T}_{\text{c}}-\tilde{T})+\gamma.$
(45)

The fitting coefficients are listed in Table 2. From the fitting results, we
observe that the coefficient $\beta$ is much near -2 of relative deviation
within $1\%$, which suggests

$\displaystyle R_{\text{N}}\sim(\tilde{T}_{\text{c}}-\tilde{T})^{-2}.$ (46)

So $R_{\text{N}}$ has a critical exponent 2. Actually, this critical exponent
can also be obtained by expanding the equation of state near the critical
point.

$\alpha$ |  $\beta$(SBH) |  $\beta$(LBH) |  $\gamma$(SBH) |  $\gamma$(LBH)
---|---|---|---|---
0.4 | -2.0342 | -1.9609 | -7.2129 | -6.3822
1.4 | -2.0131 | -1.9774 | -8.0660 | -7.6425
Table 2: Fitting coefficients $\beta$ and $\gamma$ along the coexistence small
and large black holes.

In summary, we studied the thermodynamics and Ruppeiner geometry for the five-
dimensional charged AdS black holes. Different from the neutral case, we
observe that the repulsive interaction could exist for the small black hole
with high temperature. Meanwhile, along the coexistence small or large black
hole curve, $R_{\text{N}}$ takes different values indicating that the
microscopic interaction changes among the black hole phase transition, where
the microstructure changes either. The similar thing is that the normalized
scalar curvature goes to negative infinity at the critical point, and it has a
critical exponent 2.

##  IV Phase diagram and microstructure in more than five dimensions

Next, we consider the case with the dimension number $d>5$ and confirm whether
the similar properties of $d$=5 hold.

The equation of state and the Gibbs free energy are

$\displaystyle\tilde{P}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{\text{h}}^{4-2d}}{8\pi}-\frac{(d-2)\left(d-4\pi\tilde{T}\tilde{r}_{\text{h}}-3\right)}{16\pi\tilde{r}_{\text{h}}^{2}}-\frac{\tilde{\alpha}(d-2)\left(d-8\pi\tilde{T}\tilde{r}_{\text{h}}-5\right)}{16\pi\tilde{r}_{\text{h}}^{4}},$
(47) $\displaystyle\tilde{G}$ $\displaystyle=$
$\displaystyle\frac{(d-2)\omega_{d-2}\tilde{r}_{\text{h}}^{3-d}}{4\pi\left(d^{2}-4d+3\right)}+\frac{\omega_{d-2}\tilde{r}_{\text{h}}^{d-3}\left(d-2\pi\tilde{T}\tilde{r}_{\text{h}}-2\right)}{8\pi(d-1)}+\frac{\tilde{\alpha}(d-2)\omega_{d-2}\tilde{r}_{\text{h}}^{d-5}\left(d-6\pi\tilde{T}\tilde{r}_{\text{h}}-4\right)}{4\pi(d-4)(d-1)}.$
(48)

Solving
$(\partial_{\tilde{r}_{\text{h}}}\tilde{P})_{\tilde{T},\tilde{\alpha}}=(\partial_{\tilde{r}_{\text{h}},\tilde{r}_{\text{h}}}\tilde{P})_{\tilde{T},\tilde{\alpha}}=0$,
we can obtain the critical point for different $d$. When $d$=6, we plot the
critical temperature $\tilde{T}_{\text{c}}$ as a function of $\tilde{\alpha}$
in Fig. 13. Interestingly, for fixed $\tilde{\alpha}$, more than one critical
temperatures are observed, which implies that there are several critical
points. In a small region $\tilde{\alpha}\in(2.9,3.6)$ shown in Fig. 13, we
observe three critical temperatures. So there must be some phase transitions
beyond the small-large black holes of the VdW type. Actually, there exists a
triple point for the six-dimensional charged AdS black holes, which was first
observed in our previous paper Wei2 . The exact phase diagram for the triple
point was also given. Here the triple point is out of our consideration in
this paper, so we will leave it for further study.

Figure 13: Critical temperature $\tilde{T}_{\text{c}}$ as a function of
$\tilde{\alpha}$ for $d$=6. (a) large parameter $\tilde{\alpha}$ range and (b)
small parameter $\tilde{\alpha}$ range. Figure 14: Critical temperature
$\tilde{T}_{\text{c}}$ as a function of $\tilde{\alpha}$ for $d$=7, 8, 9 from
bottom to top.

For $d$=7-9, we describe the corresponding critical temperature in Fig. 14.
For each $d$, we find that there is only one critical temperature for fixed
$\tilde{\alpha}$. Its value increases with $d$. So in higher dimensions, there
exists only the VdW type phase transition. Another interesting phenomenon is
that there always exists a lower bound for the critical temperature for each
$d$. The locations of the lower bounds ($\tilde{\alpha}$,
$\tilde{T}_{\text{c}}$) for $d$=7, 8, 9 are (5.7207, 0.1131), (5.6586,
0.1706), and (5.9334, 0.2277). Obviously, the value of $\tilde{T}_{\text{c}}$
increases with spacetime dimension number $d$. For $\tilde{\alpha}$=0, the
critical temperatures keep finite values, which are related to that of the
$d$-dimensional charged AdS black holes given in WeiWeiWei , where the small-
large black hole phase transition was found.

The normalized scalar curvature for any $d$ can be calculated through (27),
which is

$\displaystyle R_{\text{N}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{2\pi^{2}\tilde{T}^{2}\tilde{r}_{\text{h}}^{4d+2}\left(6\tilde{\alpha}+\tilde{r}_{\text{h}}^{2}\right)^{2}}{\left(\tilde{r}_{\text{h}}^{2d}\left(\tilde{r}_{\text{h}}^{2}\left(-d+2\pi\tilde{T}\tilde{r}_{\text{h}}+3\right)-2\tilde{\alpha}\left(d-6\pi\tilde{T}\tilde{r}_{\text{h}}-5\right)\right)+2\tilde{r}_{\text{h}}^{8}\right)^{2}}.$
(49)

It is easy to check that $R_{\text{N}}$ diverges at the spinodal curve. On the
other hand, it vanishes when one of the following conditions is satisfied.

$\displaystyle\tilde{r}_{\text{h}}^{2d-8}\left(\tilde{r}_{\text{h}}^{2}\left(-d+4\pi\tilde{T}\tilde{r}_{\text{h}}+3\right)-2\tilde{\alpha}\left(d-12\pi\tilde{T}\tilde{r}_{\text{h}}-5\right)\right)+2=0,$
(50)
$\displaystyle\tilde{r}_{\text{h}}^{2d-8}\left(2\tilde{\alpha}(d-5)+(d-3)\tilde{r}_{\text{h}}^{2}\right)-2=0.$
(51)

The first condition depends on $\tilde{T}$, while the second one does not. So
in the $\tilde{T}-\tilde{r}_{\text{h}}$ diagram, the first condition gives a
curve

$\displaystyle\tilde{T}=\frac{2\tilde{\alpha}(d-5)-2\tilde{r}_{\text{h}}^{8-2d}+(d-3)\tilde{r}_{\text{h}}^{2}}{4\pi\tilde{r}_{\text{h}}\left(6\tilde{\alpha}+\tilde{r}_{\text{h}}^{2}\right)},$
(52)

closely depending on $\tilde{\alpha}$ and $d$. While the second condition
shows a specific $\tilde{r}_{\text{h}}$ for each $d$, which is a vertical line
in the $\tilde{T}-\tilde{r}_{\text{h}}$ diagram. When fixing
$\tilde{\alpha}$=1, we have $\tilde{r}_{\text{h}}$=0.8180, 0.8242, and 0.8347
for $d$=7, 8, and 9, respectively.

For simplicity, we take $\tilde{\alpha}$=1 for following study of the phase
diagram and Ruppeiner geometry. The critical points are
($\tilde{P}_{\text{c}}$, $\tilde{T}_{\text{c}}$,
$\tilde{r}_{\text{hc}}$)=(0.0184, 0.1105, 1.9138), (0.0448, 0.1653, 1.3999),
(0.0954, 0.2271, 1.2171), and (0.1710, 0.2918, 1.1458) for $d$=6-9. The phase
diagrams are listed in Fig. 15. We observe that it is similar with that of
$d$=5. So for the small black hole with high temperature, $R_{\text{N}}$ is
positive, which indicates that the repulsive interaction can exist among the
microstructure. This property is different from the neutral case, while
similar to that of four-dimensional charged case WeiWeiWei .

Figure 15: Phase diagrams for $d$=6, 7, 8, and 9 with $\tilde{\alpha}$=1. The
coexistence curves and spinodal curves are marked with the red solid lines and
blue dashed lines. The sign-changing curves of $R_{\text{N}}$ is marked with
the black dot dashed lines. $R_{\text{N}}$ is positive in the shadow regions,
otherwise, it is negative. (a) $d$=6. (b) $d$=7. (c) $d$=8. (d) $d$=9.

Now, we aim to uncover the critical phenomena of the radius of the horizon and
the normalized scalar curvatures. We adopt the fitting forms (40) and (45). By
making use of the numerical calculations, we list the fitting coefficients in
Table 3. It is clear that for the coexistence small or large black hole, $b$
and $\beta$ are around 1/2 and -2, respectively. So we can conclude that

$\displaystyle\tilde{r}_{\text{h1,2}}\propto(\tilde{T}_{\text{c}}-\tilde{T})^{\frac{1}{2}},$
(53) $\displaystyle R_{\text{N}}\propto(\tilde{T}_{\text{c}}-\tilde{T})^{-2}$
(54)

hold for higher-dimensional black holes. Therefore, based on this result,
$\tilde{r}_{\text{h1,2}}$ can act as the order parameter to describe the black
hole phase transition. And the normalized scalar curvature also has a critical
exponent in higher-dimensional cases.

$d$ | $a$ | $b$ | $\beta$ | $\gamma$
---|---|---|---|---
6 | 13.7123/18.2657 | 0.4891/0.5110 | -1.9651/-2.0386 | -6.0817/-6.9303
7 | 6.5117/10.1555 | 0.4829/0.5180 | -1.9683/-2.0385 | -5.3804/-6.0545
8 | 3.2645/5.1765 | 0.4811/0.5203 | -1.9894/-2.0137 | -5.0189/-5.0989
9 | 2.1115/3.5098 | 0.4780/0.5241 | -1.9608/-2.0487 | -4.2470/-4.9299
Table 3: Fitting coefficients for $d$=6-9 with $\tilde{\alpha}$=1. …/… means
the values are for coexistence small/large black holes. The fitting forms are
given in (40) and (45).

##  V Conclusions and discussions

We have studied the phase diagram and Ruppeiner geometry for the
$d$-dimensional charged GB-AdS black holes. The VdW like phase transitions
were examined and the normalized scalar curvatures were calculated.

For $d$=5, we obtained an analytical form for the critical point of the phase
transition. After reducing these thermodynamic quantities with the black hole
charge, we found that the critical point is a curve starting at a finite point
and ending at the origin with the increase of $\tilde{\alpha}$ in the
$\tilde{P}$-$\tilde{T}$ diagram. Across the curve, the heat capacity exhibits
a $\lambda$-like phase transition behavior of 4He, which indicates that the
phase transition is second order. The phase diagrams were clearly shown in the
$\tilde{P}$-$\tilde{T}$ and $\tilde{T}$-$\tilde{r}_{\text{h}}$ diagrams.
Moreover, we also observed that the radius of the event horizon can serve as
the order parameter to characterize the black hole phase transition.

Then we constructed the Ruppeiner geometry for the five-dimensional charged
GB-AdS black holes. The corresponding normalized scalar curvatures are
calculated. After excluding the coexistence region, we observed that
$R_{\text{N}}$ can be negative or positive. According to the empirical
observation of the Ruppeiner geometry, the repulsive interaction is allowed
for the microstructure of the small black hole with higher temperature. This
result is similar to the four-dimensional charged GB-AdS black holes, while
different from the five-dimensional neutral GB-AdS black holes.

For the five-dimensional neutral GB-AdS black holes, one intriguing property
is that its microscopic interaction keeps unchanged when the phase transition
takes place. However, for the charged cases, when the system goes along the
coexistence small and large black holes, we found that their scalar curvatures
do not coincide indicating that the microscopic interaction changes during the
phase transition, where the black hole microstructure suddenly changes. So the
intriguing property does not hold for the charged black holes. We further
confirmed that there are the critical phenomena for the normalized scalar
curvatures.

Furthermore, we found that for the black holes in more than five dimensions,
they share the similar patterns of the phase diagrams and normalized scalar
curvatures. For the small black hole with higher temperature, the repulsive
interaction always dominates among their microstructure. Meanwhile, the
microscopic interaction changes when the small-large black hole phase
transition occurs, quite different from the five-dimensional neutral case.

The study provides us with preliminary properties of the microstructure of the
charged black holes in the GB gravity. Next, we will focus on the six-
dimensional charged GB-AdS black hole, where a triple point is present. These
shall help us to get deep understanding of the GB gravity.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grants No. 12075103, No. 11675064, No. 11875151, and No. 12047501), the 111
Project (Grant No. B20063), and the Fundamental Research Funds for the Central
Universities (No. Lzujbky-2019-ct06).

## References

  * (1) D. Kastor, S. Ray, and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav. 26, 195011 (2009), [arXiv:0904.2765[hep-th]].
  * (2) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60, 064018 (1999), [arXiv:hep-th/9902170].
  * (3) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes, Phys. Rev. D 60, 104026 (1999), [arXiv:hep-th/9904197].
  * (4) M. M. Caldarelli, G. Cognola, and D. Klemm, Thermodynamics of Kerr-Newman-AdS Black Holes and Conformal Field Theories, Class. Quant. Grav. 17, 399 (2000), [arXiv:hep-th/9908022].
  * (5) D. Kubiznak and R. B. Mann, $P$-$V$ criticality of charged AdS black holes, J. High Energy Phys. 1207, 033 (2012), [arXiv:1205.0559[hep-th]].
  * (6) N. Altamirano, D. Kubiznak, and R. B. Mann, Reentrant phase transitions in rotating ads black holes, Phys. Rev. D 88, 101502(R) (2013), [arXiv:1306.5756[hep-th]].
  * (7) N. Altamirano, D. Kubiznak, R. B. Mann, and Z. Sherkatghanad, Kerr-AdS analogue of triple point and solid/liquid/gas phase transition, Class. Quant. Grav. 31, 042001 (2014), [arXiv:1308.2672[hep-th]].
  * (8) N. Altamirano, D. Kubiznak, R. B. Mann, and Z. Sherkatghanad, Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume, Galaxies 2, 89 (2014), [arXiv:1401.2586[hep-th]].
  * (9) B. P. Dolan, A. Kostouki, D. Kubiznak, and R. B. Mann, Isolated critical point from Lovelock gravity, Class. Quant. Grav. 31, 242001 (2014), [arXiv:1407.4783[hep-th]].
  * (10) S.-W. Wei and Y.-X. Liu, Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes, Phys. Rev. D 87, 044014 (2013), [arXiv:1209.1707[gr-qc]].
  * (11) S.-W. Wei and Y.-X. Liu, Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space, Phys. Rev. D 90, 044057 (2014), [arXiv:1402.2837[hep-th]].
  * (12) A. M. Frassino, D. Kubiznak, R. B. Mann, and F. Simovic, Multiple reentrant phase transitions and triple points in Lovelock thermodynamics, J. High Energy Phys. 1409, 080 (2014), [arXiv:1406.7015[hep-th]].
  * (13) R.-G. Cai, L.-M. Cao, L. Li, and R.-Q. Yang, $P$-$V$ criticality in the extended phase space of Gauss-Bonnet black holes in AdS space, J. High Energy Phys. 1309, 005 (2013), [arXiv:1306.6233[gr-qc]].
  * (14) H. Xu, W. Xu, and L. Zhao, Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions, Eur. Phys. J. C 74, 3074 (2014), [arXiv:1405.4143[gr-qc]].
  * (15) R. A. Hennigar, W. G. Brenna, and R. B. Mann, $P$-$V$ criticality in quasitopological gravity, J. High Energy Phys. 1507, 077 (2015), [arXiv:1505.05517[hep-th]].
  * (16) R. A. Hennigar and R. B. Mann, Reentrant phase transitions and van der Waals behaviour for hairy black holes, Entropy 17, 8056 (2015), [arXiv:1509.06798[hep-th]].
  * (17) R. A. Hennigar, R. B. Mann, and E. Tjoa, Superfluid black holes, Phys. Rev. Lett. 118, 021301 (2017), [arXiv:1609.02564[hep-th]].
  * (18) D.-C. Zou, R. Yue, and M. Zhang, Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity, Eur. Phys. J. C 77, 256 (2017), [arXiv:1612.08056[gr-qc]].
  * (19) S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  * (20) J. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4, 737 (1972).
  * (21) J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  * (22) J. M. Bardeen, B. Carter, and S. Hawking, The four laws of black hole mechanics, Commun. Math. Phys.31, 161 (1973).
  * (23) G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67, 605 (1995); Erratum: Rev. Mod. Phys. 68, 313 (1996).
  * (24) R.-G. Cai and J.-H. Cho, Thermodynamic curvature of the BTZ black hole, Phys. Rev. D 60, 067502 (1999).
  * (25) S.-W. Wei and Y.-X. Liu, Insight into the microscopic structure of an AdS black hole from thermodynamical phase transition, Phys. Rev. Lett. 115, 111302 (2015), [arXiv:1502.00386[gr-qc]]; Erratum: Phys. Rev. Lett. 116, 169903 (2016).
  * (26) S.-W. Wei, Y.-X. Liu, and R. B. Mann, Repulsive Interactions and Universal Properties of Charged AdS Black Hole Microstructures, Phys. Rev. Lett. 123, 071103 (2019), [arXiv:1906.10840[gr-qc]].
  * (27) S.-W. Wei, Y.-X. Liu, and R. B. Mann, Ruppeiner Geometry, Phase Transitions, and the Microstructure of Charged AdS Black Holes, Phys. Rev. D 100, 124033 (2019) [arXiv:1909.03887[gr-qc]].
  * (28) A. Dehyadegari, A. Sheykhi, and A. Montakhab, Critical behaviour and microscopic structure of charged AdS black holes via an alternative phase space, Phys. Lett. B 768, 235 (2017), [arXiv:1607.05333[gr-qc]].
  * (29) M. K. Zangeneh, A. Dehyadegari, M. R. Mehdizadeh, B. Wang, and A. Sheykhi, Thermodynamics, phase transitions and Ruppeiner geometry for Einstein-dilaton Lifshitz black holes in the presence of Maxwell and Born-Infeld electrodynamics, Eur. Phys. J. C 77, 423 (2017), [arXiv:1610.06352[hep-th]].
  * (30) M. Chabab, H. E. Moumni, S. Iraoui, K. Masmar, and S. Zhizeh, More insight into microscopic properties of RN-AdS black hole surrounded by quintessence via an alternative extended phase space, Int. J. Geom. Meth. Mod. Phys. 15, 1850171 (2018), [arXiv:1704.07720[gr-qc]].
  * (31) Y.-G. Miao and Z.-M. Xu, Microscopic structures and thermal stability of black holes conformally coupled to scalar fields in five dimensions, Nucl. Phys. B 942, 205 (2019), [arXiv:1711.01757[hep-th]].
  * (32) Y.-Z. Du, R. Zhao, and L.-C. Zhang, Microstructure and continuous phase transition of the gauss-bonnet ads black hole, Adv. High Energy Phys. 2020, 6395747 (2020), [arXiv:1901.07932[hep-th]].
  * (33) Z.-M. Xu, B. Wu, and W.-L. Yang, The fine micro-thermal structures for the Reissner-Nordstrom black hole, Chin. Phys. C 44, 095106 (2020), [arXiv:1910.03378[gr-qc]].
  * (34) A. Ghosh and C. Bhamidipati, Thermodynamic geometry for charged Gauss-Bonnet black holes in AdS spacetimes, Phys. Rev. D 101, 046005 (2020) [arXiv:1911.06280[gr-qc]].
  * (35) A. N. Kumara, C. A. Rizwan, K. Hegde, and Ajith K. M, Repulsive Interactions in the Microstructure of Regular Hayward Black Hole in Anti-de Sitter Spacetime, Phys. Lett. B 807, 135556 (2020), [arXiv:2003.10175[gr-qc]].
  * (36) P. K. Yerra and C. Bhamidipati, Ruppeiner Geometry, Phase Transitions and Microstructures of Black Holes in Massive Gravity, Int. J. Mod. Phys. A 35, 2050120 (2020), [arXiv:2006.07775[hep-th]].
  * (37) B. Wu, C. Wang, Z.-M. Xu, and W.-L. Yang, Ruppeiner geometry and thermodynamic phase transition of the black hole in massive gravity, [arXiv:2006.09021[gr-qc]].
  * (38) C. L. A. Rizwan, A. N. Kumara, K. Hegde, and D. Vaid, Coexistent Physics and Microstructure of the Regular Bardeen Black Hole in Anti-de Sitter Spacetime, Annals Phys. 422, 168320 (2020), [arXiv:2008.06472[gr-qc]].
  * (39) A. N. Kumara, C.L. A. Rizwan, K. Hegde, M. S. Ali, and Ajith K.M, Ruppeiner Geometry, Reentrant Phase transition and Microstructure of Born-Infeld AdS Black Hole, Phys. Rev. D 103, 044025 (2021), [arXiv:2007.07861[gr-qc]].
  * (40) S. A. H. Mansoori, M. Rafiee, and S.-W. Wei, Universal criticality of thermodynamic curvatures for charged AdS black holes, Phys. Rev. D 102, 124066 (2020), [arXiv:2007.03255[gr-qc]].
  * (41) A. N. Kumara, C.L. A. Rizwan, K. Hegde, M. S. Ali, and Ajith K.M, Microstructure of five-dimensional neutral Gauss-Bonnet black hole in anti-de Sitter spacetime via $P$-$V$ criticality, [arXiv:2006.13907[gr-qc]].
  * (42) S.-W. Wei, Y.-X. Liu, and R. B. Mann, Novel dual relation and constant in Hawking-Page phase transition, Phys. Rev. D 102, 104011 (2020), [arXiv:2006.11503[gr-qc]].
  * (43) A. Dehyadegari, A. Sheykhi, and S.-W. Wei, Microstructure of charged AdS black hole via $P$-$V$ criticality, Phys. Rev. D 102, 104013 (2020), [arXiv:2006.12265[gr-qc]].
  * (44) S.-W. Wei and Y.-X. Liu, Extended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space, Phys. Rev. D 101, 104018 (2020), [arXiv:2003.14275[gr-qc]].
  * (45) S.-W. Wei and Y.-X. Liu, Intriguing microstructures of five-dimensional neutral Gauss-Bonnet AdS black hole, Phys. Lett. B 803, 135287 (2020), [arXiv:1910.04528[gr-qc]].
  * (46) R. Zhou, Y.-X. Liu, and S.-W. Wei, Phase transition and microstructures of five-dimensional charged Gauss-Bonnet-AdS black holes in the grand canonical ensemble, Phys. Rev. D 102, 124015 (2020), [arXiv:2008.08301[gr-qc]].
  * (47) Y.-P. Hu, L. Cai, X. Liang, and H. Zhang, Divergence Behavior of Thermodynamic Curvature Scalar at Critical Point in the extended Phase Space of generic Black Holes, [arXiv:2010.09363[gr-qc]].
  * (48) Z.-M. Xu, Analytic phase structures and thermodynamic curvature for the charged AdS black hole in alternative phase space, Front. Phys. 16, 24502 (2021), [arXiv:2011.06736[gr-qc]].
  * (49) J.-X. Mo and G.-Q. Li, Coexistence curves and molecule number densities of AdS black holes in the reduced parameter space, Phys. Rev. D 92, 024055 (2015), [arXiv:1604.07931[gr-qc]].
  * (50) D. G. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett. 55, 2656 (1985).
  * (51) R. G. Cai, Gauss-Bonnet Black Holes in AdS Spaces, Phys. Rev. D 65, 084014 (2002), [arXiv:hep-th/0109133].
  * (52) D. L. Wiltshire, Spherically Symmetric Solutions Of Einstein-maxwell Theory With A Gauss-bonnet Term, Phys. Lett. B 169, 36 (1986).
  * (53) M. Cvetic, S. Nojiri, and S. D. Odintsov, Black Hole Thermodynamics and Negative Entropy in deSitter and Anti-deSitter Einstein-Gauss-Bonnet gravity, Nucl. Phys. B 628, 295 (2002), [arXiv:hep-th/0112045].
  * (54) S.-W. Wei and Y.-X. Liu, Clapeyron equations and fitting formula of the coexistence curve in the extended phase space of charged AdS black holes, Phys. Rev. D 91, 044018 (2015), [arXiv:1411.5749[hep-th]].
  * (55) S.-W. Wei, P. Cheng, and Y.-X. Liu, Analytical and exact critical phenomena of $d$-dimensional singly spinning Kerr-AdS black holes, Phys. Rev. D 93, 084015 (2016), [arXiv:1510.00085[gr-qc]].