††institutetext: Department Of Physics,
Dibrugarh University, Dibrugarh,786004, Assam, India††institutetext:
Theoritical Physics Division, Center for Atmospheric Studies, Dibrugarh
University

# Holographic CFT thermodynamics of charged, rotating black holes in $D=4$
dimension

Abhishek Baruah  Prabwal Phukon<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the holographic thermodynamics of $4-D$ Kerr-Newman AdS black holes.
We consider the conformal thermal states dual to KN AdS black holes and work
out the corresponding thermodynamics in 10 ensembles. These ensembles are:
fixed $(\mathcal{Q},\mathcal{J},\mathcal{V},C)$, fixed
$(\mathcal{Q},\Omega,\mathcal{V},C)$, fixed $(\varphi,\Omega,\mathcal{V},C)$,
fixed $(\varphi,\mathcal{J},\mathcal{V},C)$, fixed
$(\mathcal{Q},\mathcal{J},p,C)$, fixed $(\mathcal{Q},\Omega,p,C)$,fixed
$(\varphi,\mathcal{J},p,C)$,fixed $(\mathcal{Q},\mathcal{J},p,\mu)$ and fixed
$(\varphi,\Omega,p,\mu)$ ensembles. Here $\varphi$, $\mathcal{Q}$, $\Omega$,
$\mathcal{J}$, $p$, $\mathcal{V}$, $C$ and $\mu$ denotes the electric
potential, electric charge, angular velocity, angular momentum, CFT pressure,
CFT volume, central charge, and chemical potential respectively. In the fixed
$(\mathcal{Q},\mathcal{J},\mathcal{V},C)$ ensemble, we observe a first order
phase transition for $\mathcal{Q}<\mathcal{Q}_{crit}$,
$\mathcal{J}<\mathcal{J}_{crit}$ and $C>C_{crit}$. In the fixed
$(\mathcal{Q},\Omega,\mathcal{V},C)$ ensemble, we again find a first-order
phase transition for $\mathcal{Q}<\mathcal{Q}_{crit}$, $\Omega<\Omega_{crit}$
and $C>C_{crit}$. The fixed $(\varphi,\Omega,\mathcal{V},C)$ ensemble is
characterized by a confinement/de-confinement phase transition. In the fixed
$(\varphi,\mathcal{J},\mathcal{V},C)$ ensemble, we see a first order phase
transition for $\mathcal{J}<\mathcal{J}_{crit}$, $\varphi<\varphi_{crit}$ and
$C>C_{crit}$. Finally, in the fixed
$(\mathcal{Q},\mathcal{J},p,C)$,$(\mathcal{Q},\Omega,p,C)$,$(\varphi,\mathcal{J},p,C)$,$(\mathcal{Q},\mathcal{J},p,\mu)$,$(\varphi,\Omega,p,\mu)$
ensembles, we do not observe any critical behavior or phase transition.

##  1 Introduction

Understanding the underlying similarities between the laws of black hole
mechanics and thermodynamics about half a century back culminated in the
formulation of what is now known as black hole thermodynamics a1 ; a2 ; a3 ; a
; b ; a4 ; b1 ; b2 . The two key equations of black hole thermodynamics are
those involving Hawking’s temperature $T$ and Bekenstein-Hawking entropy $S$.
These are as follows:

$T=\frac{\kappa}{2\pi},\quad S=\frac{A}{4G_{N}}$ (1)

Here. $\kappa$, $A$, and $G_{N}$ are surface gravity, horizon area, and
Newton’s universal gravitational constant respectively.

One particular class of black holes that have contributed immensely to the
subsequent development and expansion of conventional black hole thermodynamics
is the asymptotically anti-de-Sitter (AdS) black holes. AdS black holes,
depending on their nature (uncharged, charged, rotating, hairy) and the
ensemble under consideration, are characterized by a diverse range of rich
phase structuresb3 ; b4 ; b5 ; j ; d ; k . At present, apart from the
traditional thermodynamic methods, several alternative tools like
thermodynamic geometryaf1 ; af2 ; af3 ; af4 ; af5 ; af6 ; af7 ; af8  and
thermodynamic topologyg1 ; g2 ; g3 ; g4 ; g5 ; g6  are used to gain useful
insights into these phase structures.

From the traditional black hole thermodynamics, next comes the Extended Phase
space thermodynamics (EPST). This concept was first given by Kastor, Ray,
Traschen, and later more insights and development were done by other
researchers. This EPST suggested the introduction of a new pair of
thermodynamic variables, namely the negative cosmological constant which is
linked to the pressure, treating it as a state variable and the conjugate
variable, namely the thermodynamic volume.f ; g ; h ; i . The thermodynamic
pressure is given as

$P=-\frac{\Lambda}{8\pi
G_{N}},\quad\text{with}\quad\Lambda=-\frac{d(d-1)}{2L^{2}}$ (2)

where $L$ is given as the AdS curvature radius, $G_{N}$ is the Newton’s
constant, $d$ denotes the number of bulk spacetime dimensions and $V=-\Theta$
is identified as the thermodynamic volume in the extended first law of black
holes. But in this formalism, a slightly odd way to introduce the mass
parameter as the enthalpy rather than the internal energy.

From the holographic viewpoint, it is quite difficult to grasp the idea of the
dimensional dependant factors and the $\Lambda$ term in the generalized Smarr
relation formula. The AdS/CFT correspondence, which is also known as the
gauge/ gravity duality says that the thermodynamics of the black holes in the
"bulk" space-time corresponds to the thermodynamics of the large-N strongly
coupled gauge theories living on the bulk space-time asymptotic boundary c ;
c1 . The Hawking-Phase transition, observed between the thermal radiation and
a large AdS black hole is one of the foremost examples, which can be explained
as the confinement/deconfinement phase transition of the gauge fields. We can
say that the thermodynamic variables in (1) for the black holes correspond
with the energy $E$, entropy $S$, and the temperature $T$ of the thermal
states in the boundary theory. The generalized Smarr relation which relates
all these variables listed above in the gravity theory should have a one-to-
one correspondence to the Euler relation for the field theory thermodynamic
quantity. But sadly, the pressure explanation of the $\Lambda$ term in the
Smarr formula does not directly carry to the field theory. Hence the bulk
pressure $P$ matches with the boundary CFT pressure $p$ and the bulk
thermodynamical volume is also not related to the spatial volume $\mathcal{V}$
of the CFT boundary.

The study of thermal equilibrium of AdS black holes has achieved a variety of
interesting behaviors such as the first order phase transition to the
radiation which is analogous to the confinement/deconfinement of the quark-
gluon plasmae , also for charged AdS black that undergoes a Van der Waals type
phase transition l ; m  and also for rotating black holes n ; o ; p . The
study of black hole thermodynamics q  garnered study in multiple fields such
as polymer transitions,m1  superfluid transitionsr , triple pointsr1 ; r2  and
reentrant phase transitions s ; s1 , micro-structures of black hole t  and
multi-critical phase transitions u ; v . In a recent study, Visser bb  gave a
set of thermodynamic variables to this formalism that is referred to as
central charge $C$ and the chemical potential $\mu$. With this extension, the
study of the central charge criticality has come to existence cc1 .

The more potent phase behavior of the AdS black hole is due to the
incorporation of the negative cosmological constant $\Lambda$, as an
additional thermodynamic variable leading to the existence of new phase
transitions. Also, the variation of Newton’s constant has been added to the
EPST formalismw ; aa1 . Newton’s constant is used as a ’book-keeping’ device
for finding the correct thermodynamical interpretation for the black holes
extended first law. The extended first law and the generalized Smarr relation
in $D=d+1$ dimensions can be given as

$dM=TdS+\Phi dQ+\Omega dJ+VdP$ (3) $M=\frac{d-1}{d-2}(TS+\Omega J)+\Phi
Q-\frac{2}{d-2}PV$ (4)

Now if we let the variation of Newton’s constant be allowed, then the extended
first law can be rewritten as:-

$dM=\frac{\kappa}{8\pi}d\left(\frac{A}{G_{N}}\right)+\Phi dQ+\Omega
dJ+\frac{\Theta}{8\pi}d\left(\frac{\Lambda}{G_{N}}\right)-\left(M-\frac{\kappa
A}{8\pi G_{N}}-\Phi Q-\Omega J-\frac{\Theta\Lambda}{8\pi
G_{N}}\right)\frac{dG_{N}}{G_{N}}$ (5)

We can interpret the first and the fourth terms as the $TdS$ term and the
$VdP$ term. But the thermodynamical interpretation of the final term cannot be
identified and it cannot be put to zero by using the Smarr relation
In recent years the study of holographic explanation of black holes has been
studied intensively x ; x1 ; y ; z . In many works, it is said that when we
vary $\Lambda$, it is related to varying the number of colors $N$, or the
number of degrees of freedom $N^{2}$, in the boundary field theory. Varying
the $N$ will in turn correspond to vary the number of branes for the gauge
theories arising from the coincident D-branes. Following in the conformal
field theories (CFTs), the central charge $C$ is given by the number of
degrees of freedom, which when varied takes us from one CFT to another. The
holographic CFT which also dual to the Einstien gravity, the central charge
$C$ corresponds to $C\sim L^{d-1}/G$, where $G$ is Newton’s constant in $d+1$
dimensions and $L$ is the radius of curvature of the bulk geometry, which
relates to $\Lambda$ by $\Lambda=-d(d-1)/2L^{2}$. Therefore, varying $G$ and
$\Lambda$ in the bulk theory is dual to varying $C$ in the boundary CFT. In aa
it is seen that when we vary $\Lambda$ in the bulk CFT not only corresponds to
a change in the $C$ (or $N$), but it also suggests a change in the volume $V$
of the spatial boundary. The reason is that the boundary curvature radius of a
particular boundary metric is equal to the bulk curvature radius $L$. We also
see that the electric potential and its analogous CFT electric charge are
related to the AdS length scale. Hence we cannot directly project the first
law thermodynamics of the bulk spacetime into the holographic dual boundary
theory. To solve this projection dissimilarity issue the researchers have
given out a theory to maintain the CFT’s central charge fixed by dynamically
varying the CFT volume $\mathcal{V}$ and the CFT central charge $C$ without
introducing the variable Newton’s constantbb1 ; bb2 ; bb3 ; cc . The bulk AdS
space-time conformal completion is given as:-

$ds^{2}=\omega(-dt^{2}+l^{2}d\Omega^{2}_{d-2})$ (6)

where the dimensionless arbitrary conformal factor is $\omega$ which reflects
the conformal symmetry of the boundary theory. $d\Omega^{2}_{d-2}$ is
considered as a metric on a unit $(d-2)$\- dimensional sphere having the
corresponding volume as $\Omega_{d-2}$. By taking $\omega$ as an independent
coordinate, the CFT volume reads as:-

$\mathcal{V}=\Omega_{d-2}R^{d-2}$ (7)

where $R=\omega l$ is given as the curvature radius which is considered a
variable on the plane where the CFT resides. The expression of the central
charge $C$ to which the CFT volume $\mathcal{V}$ is independent is given as

$C=\frac{\Omega_{d-2}L^{d-2}}{16\pi G_{N}}$ (8)

Using the AdS CFT holographic dictionary

$E=\frac{M}{\omega},\quad\mathcal{T}=\frac{T}{\omega},\quad\mathcal{S}=S,\quad\mathcal{J}=J,\quad\varphi=\frac{\Phi}{\omega
l},\quad\mathcal{Q}=Ql$ (9)

Using the dictionary given above the bulk first law is given as

$dE=\mathcal{T}\mathcal{S}\varphi d\mathcal{Q}-p\mathcal{V}+\mu dC+\Omega
d\mathcal{J}$ (10)

and the Smarr relation is given as

$E=\mathcal{T}\mathcal{S}+\varphi\mathcal{Q}+\mu C+\Omega J$ (11)

followed by the two expressions for the chemical potential $\mu$ and for the
CFT pressure $p$ respectively is given as:-

$\mu=\frac{1}{C}(E-\mathcal{T}\mathcal{S}-\Omega J-\varphi Q),\quad
p=\frac{E}{(d-2)\mathcal{V}}$ (12)

The analogous dual CFT does not change even if we see rescaling in the bulk
cosmological constant. As a consequence, we can calculate the first law of the
black hole thermodynamics, from the dual boundary field theory. In this way,
it explains a holographic explanation of the black hole chemistry, ideally for
the Reissner-Nordstrom AdS black holes Van der Waals phase transition. This
phase change is determined by the analogous field theory degrees of freedom,
which agrees with the central charge criticality examinations for the Gauss-
Bonnet black holes gg ; hh  and for non-linear electromagnetic black holesee ;
ff . In the analogous CFT description, these black holes exhibit zeroth-order,
first-ordered, and second-ordered phase transitions but upon careful
examination, it was revealed that there is no criticality in the pressure-
volume CFT, which verifies that the CFT states which is regarded as a dual to
the charged AdS black holes in not a Van der Waals fluid.

The main aim of this paper is to further generalize or we can say incorporate
together previous examinations of the holographic thermodynamics namely the
CFT thermodynamics of charged dd  and the CFT thermodynamics of rotating black
holes ii ; dd1 . We can divide it into two main ideas. Firstly, we review the
previously studied generalized mass/energy formulas for the charged and
rotating Kerr-Newman AdS black holes (KN-AdS) from both sides namely the
holographic CFT side and the bulk AdS. Secondly, taking the ensembles from the
CFT side, we investigate all the ensembles namely the 10 ensembles, and plot
the free energies of each ensemble varying all the parameters. In doing so we
can examine if some ensembles do exhibit phase transitions and exhibit
criticality in the CFT thermodynamics.
This paper is divided into 5 sections. In section 1 we have given the
introduction of the paper as well as the approach as to what we want to study
and also the motivations. In section 2, we review the mass-energy formulas
from the extended phase space to the CFT thermodynamics. In section 3, we have
the main results where study extensively the criticality or phase transitions
for various ensembles for the thermal states analogous to the 4-D KN-AdS black
hole. In section 4, we again study the phase transitions for remaining
p-ensembles, and in section 5 we give out the conclusion.

##  2 Mass energy formulas from the extended thermodynamics to the CFT
thermodynamics

In this section, we review the mass formulas for the charged and rotating
black holes ii . We do this in the extended phase space thermodynamics in
terms of the bulk event horizon area, electric charge, angular momentum, and
the cosmological constant. Then after this, we will look into the black hole
mass formula in the so-called "mixed thermodynamics" where there is a
dependence on black hole entropy, electric charge, angular momentum,
thermodynamic pressure, and the boundary CFT central charge. After all this,
we review the energy formulas of the dual CFT theory from the mass formulas of
the extended phase space thermodynamics of the KN-AdS black holes.

###  2.1 Extended Phase Space Thermodynamics (EPST)

The Einstein-Maxwell theory action is taken here as

$I=\frac{1}{16\pi G_{N}}\int d^{4}x\sqrt{-g}(R-2\Lambda-\mathcal{F}^{2})$ (13)

Here, the $\Lambda$ is regarded as the cosmological constant, $G_{N}$ as the
gravitational constant and $\mathcal{F}_{ab}$ as the strength of the $U(1)$
field, and lastly $R$ is the Ricci scalar. The solution is given below:-

$ds^{2}=-\frac{\Delta_{r}}{\Sigma^{2}}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right)^{2}+\sigma^{2}\left(\frac{dr^{2}}{\Delta_{r}}+\frac{d\theta^{2}}{H_{\theta}}\right)+\frac{H_{\theta}\sin^{2}\theta}{\Sigma^{2}}\left[adt-\frac{r^{2}+a^{2}}{\Xi}d\phi\right]^{2}$
(14) $\mathcal{F}_{ab}=(dB)_{ab}$ (15)
$B=-\frac{G_{N}qr}{\Sigma^{2}}\left[dt-a\sin^{2}\theta
d\phi\right]+\Phi_{t}dt$ (16)

where all the unknown terms are

$\begin{split}&\Delta_{r}=r^{2}-2G_{N}mr+a^{2}+G_{N}^{2}q^{2}+\frac{(r^{2}+a^{2})r^{2}}{l^{2}}\\\
&H_{\theta}=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta\\\
&\Sigma^{2}=r^{2}+a^{2}\cos^{2}\theta\\\
&\Phi_{t}=\frac{G_{N}qr_{+}}{a^{2}+r^{2}_{+}},\\\
&\Xi=1-\frac{a^{2}}{l^{2}}.\end{split}$ (17)

From the black hole when we set $\Delta_{r}=0$ we get the event horizon
$r_{+}$. Here, the gauge potential is given by $B_{a}$, where we get vanishing
gauge potential for the black holes for the values of $\Phi_{t}$. The related
parameters namely the ADM mass $m$, electric charge $q$ and the angular
momentum $J$ are given as

$M=\frac{m}{\Xi^{2}}$ (18) $J=\frac{am}{\Xi^{2}}=Ma,\quad Q=\frac{q}{\Xi}$
(19)

The cosmological constant is given as

$\Lambda=-\frac{3}{l^{2}}$ (20)

where the curvature radius or AdS length scale is given as $l$.
So, now we can state the horizon area $A$, thermodynamic pressure $P$ as

$A=\frac{4\pi(r^{2}_{+}+a^{2})}{\Xi}$ (21) $P=-\frac{\Lambda}{8\pi G_{N}}$
(22)

The temperature, angular velocity thermodynamic volume, and electric potential
are given as

$T=\frac{-l^{2}(a^{2}+G^{2}_{N}q^{2})+r^{2}_{+}(a^{2}+l^{2})+3r^{2}_{+}}{4\pi
l^{2}r_{+}(a^{2}+r^{2}_{+})}$ (23)
$\Omega_{b}=\frac{a(l^{2}+r^{2}_{+})}{l^{2}(a^{2}+r^{2}_{+})}$ (24)
$V=\frac{2\pi
l^{2}(a^{2}l^{2}(a^{2}+G^{2}_{N}q^{2})-(r^{4}_{+}(a^{2}-2l^{2}))-r^{2}_{+}(a^{4}-3a^{2}l^{2}))}{3r_{+}(a^{2}-l^{2})^{2}}$
(25) $\Phi=\frac{G_{N}qr_{+}}{a^{2}+r^{2}_{+}}$ (26)

We can relate the surface gravity $\kappa$ and the entropy $S$ from the area
$A$ and the temperature $T$ as

$S=\frac{A}{4G_{N}}=-\frac{\pi l^{2}\left(a^{2}+r^{2}\right)}{G(a-l)(a+l)}$
(27) $\kappa=2\pi T$ (28)

The KN- AdS black hole generalised mass $M$ in terms of $(G_{N},Q,J,P,S)$ is
given as

$M^{2}=\frac{12\pi^{2}J^{2}\left(8G^{2}PS+3\right)+16G^{3}PS^{2}\left(4PS^{2}+3\pi
Q^{2}\right)+G^{2}\left(48PS^{3}+9\pi^{2}Q^{4}\right)+18\pi
GQ^{2}S+9S^{2}}{36\pi GS}$

(29)

The thermodynamic first law of the black hole by using the above mass formula
in the Extended Phase Space Thermodynamics is given as

$dM=TdS+\Phi dQ+VdP+\Omega_{b}dJ$ (30)

The corresponding Smarr relation can be written as-

$M=2TS+2\Omega_{b}J+\Phi Q-2PV$ (31)

We can also write the mass formula by replacing the entropy $S$ with the area
$A$ (27) and the pressure $P$ with cosmological constant $\Lambda$ (22). If we
wish to incorporate the dynamic nature of Newton’s gravitation constant
$G_{N}$. jj  The new mass formula looks like-

$M^{2}=\frac{A^{3}\Lambda^{2}}{2304\pi^{3}G^{3}}-\frac{A^{2}\Lambda}{96\pi^{2}G^{2}}+\frac{\pi\left(G^{2}Q^{4}+4J^{2}\right)}{A}+\frac{A\left(3-2G^{2}\Lambda
Q^{2}\right)}{48\pi G^{2}}+\frac{1}{6}\left(3Q^{2}-2J^{2}\Lambda\right)$ (32)

Then the first law of thermodynamics treating $\Lambda$, $G_{N}$ as parameters
as

$dM=\frac{\kappa}{8\pi G_{N}}dA+\Phi dQ+\Omega_{b}dJ-\frac{V}{8\pi
G_{N}}d\Lambda-(M-\Phi Q-\Omega_{b}J)\frac{dG_{N}}{G_{N}}$ (33)

We can derive the related quantities as:-

$\begin{split}&\frac{\kappa}{8\pi G_{N}}=\left(\frac{\partial M}{\partial
A}\right)_{Q,J,\Lambda,G_{N}}\\\ =&\frac{16\pi^{2}A^{2}G\left(3-2G^{2}\Lambda
Q^{2}\right)-16\pi
A^{3}G\Lambda+A^{4}\Lambda^{2}-768\pi^{4}G^{3}\left(G^{2}Q^{4}+4J^{2}\right)}{32\pi^{3/2}A^{2}G^{3}\sqrt{\frac{A^{3}\Lambda^{2}}{G^{3}}-\frac{24\pi
A^{2}\Lambda}{G^{2}}+\frac{2304\pi^{4}\left(G^{2}Q^{4}+4J^{2}\right)}{A}-\frac{48\pi^{2}A\left(2G^{2}\Lambda
Q^{2}-3\right)}{G^{2}}+384\pi^{3}\left(3Q^{2}-2J^{2}\Lambda\right)}}\end{split}$
(34)

$\begin{split}&\Phi=\left(\frac{\partial M}{\partial
Q}\right)_{A,J,\Lambda,G_{N}}\\\ =&\frac{2\left(-A^{2}\Lambda Q+12\pi
AQ+48\pi^{2}G^{2}Q^{3}\right)}{A\sqrt{\frac{A^{3}\Lambda^{2}}{\pi
G^{3}}-\frac{24A^{2}\Lambda}{G^{2}}+\frac{2304\pi^{3}\left(G^{2}Q^{4}+4J^{2}\right)}{A}+48\pi
A\left(\frac{3}{G^{2}}-2\Lambda
Q^{2}\right)+384\pi^{2}\left(3Q^{2}-2J^{2}\Lambda\right)}}\end{split}$ (35)

$\begin{split}&-\frac{V}{8\pi G_{N}}=\left(\frac{\partial
M}{\partial\Lambda}\right)_{A,Q,J,G_{N}}\\\ =&\frac{-12\pi
A^{2}G+A^{3}\Lambda-48\pi^{2}AG^{3}Q^{2}-384\pi^{3}G^{3}J^{2}}{48\pi^{3/2}G^{3}\sqrt{\frac{A^{3}\Lambda^{2}}{G^{3}}-\frac{24\pi
A^{2}\Lambda}{G^{2}}+\frac{2304\pi^{4}\left(G^{2}Q^{4}+4J^{2}\right)}{A}-\frac{48\pi^{2}A\left(2G^{2}\Lambda
Q^{2}-3\right)}{G^{2}}+384\pi^{3}\left(3Q^{2}-2J^{2}\Lambda\right)}}\end{split}$
(36)

$\begin{split}&\frac{\Omega_{b}J+\Phi Q-M}{G_{N}}=\left(\frac{\partial
M}{\partial G_{N}}\right)_{A,Q,J,\Lambda}\\\ &=\frac{16\pi
A^{3}G\Lambda-96\pi^{2}A^{2}G-A^{4}\Lambda^{2}+1536\pi^{4}G^{5}Q^{4}}{32\pi^{3/2}AG^{4}\sqrt{\frac{A^{3}\Lambda^{2}}{G^{3}}-\frac{24\pi
A^{2}\Lambda}{G^{2}}+\frac{2304\pi^{4}\left(G^{2}Q^{4}+4J^{2}\right)}{A}-\frac{48\pi^{2}A\left(2G^{2}\Lambda
Q^{2}-3\right)}{G^{2}}+384\pi^{3}\left(3Q^{2}-2J^{2}\Lambda\right)}}\end{split}$
(37)

The corresponding integral Smarr relation related to the first law is (33)

$M=\frac{\kappa A}{4\pi G_{N}}+2\Omega_{b}J+\Phi Q+\frac{V\Lambda}{4\pi
G_{N}}$ (38)

###  2.2 Mixed Thermodynamics

To incorporate the boundary central charge from the thermodynamic first law,
we take into consideration the holographic dual relation between the AdS scale
$l$, Newton’s constant $G_{N}$, and the central charge $C$. which is given
as:-

$C=\frac{\Omega_{2}l^{2}}{16\pi G_{N}}$ (39)

Here, $\Omega_{2}=4\pi$. In terms of the bulk thermodynamic pressure and the
central charge, Newton’s constant $G_{N}$ can be achieved by combining (22),
(20) as

$G_{N}=\frac{1}{4}\sqrt{\frac{3}{2\pi CP}}$ (40)

Now of we incorporate the value of Newton’s gravitational constant from the
above equation (40) to the mass formula in (29) we get:-

$\begin{split}&M^{2}=\frac{\sqrt{\frac{1}{CP}}\left(24\pi^{3/2}CP\left(S\left(\sqrt{6\pi}Q^{2}\sqrt{\frac{1}{CP}}+4S\right)+16\pi^{2}J^{2}\right)+\pi^{5/2}\left(96J^{2}PS+9Q^{4}\right)\right)}{48\sqrt{6}\pi^{2}S}\\\
&+\frac{\sqrt{\frac{1}{CP}}\left(8\sqrt{6}P^{2}S^{4}\sqrt{\frac{1}{CP}}+6\pi\sqrt{6}PQ^{2}S^{2}\sqrt{\frac{1}{CP}}+48\sqrt{\pi}PS^{3}\right)}{48\sqrt{6}\pi^{2}S}\end{split}$
(41)

The thermodynamic first law corresponding to the above mass formula (41) is

$dM=TdS+\Phi dQ+V_{C}dP+\mu_{b}dC+\Omega_{b}dJ$ (42)

The pressure $P$ has a conjugate quantity the thermodynamic volume $V_{C}$ and
the central charge $C$ conjugate quantity is a chemical potential $\mu_{b}$.
The related quantities can be calculated as given below:-

$\begin{split}&T=\left(\frac{\partial M}{\partial S}\right)_{Q,P,C,J}\\\
=&\frac{\sqrt[4]{\frac{3}{2}}\sqrt{\frac{1}{CP}}\left(-32\pi^{3/2}CP\left(4\pi^{2}J^{2}-S^{2}\right)+8\sqrt{6}P^{2}S^{4}\sqrt{\frac{1}{CP}}+2\pi\sqrt{6}PQ^{2}S^{2}\sqrt{\frac{1}{CP}}+32\sqrt{\pi}PS^{3}-3\pi^{5/2}Q^{4}\right)}{8\pi
S^{3/2}\sqrt{3\pi^{5/2}\sqrt{\frac{1}{CP}}\left(32J^{2}PS+3Q^{4}\right)+\frac{96\pi^{3/2}\left(4\pi^{2}J^{2}+S^{2}\right)}{\sqrt{\frac{1}{CP}}}+\frac{2\sqrt{6}S^{2}\left(4PS^{2}+3\pi
Q^{2}\right)}{C}+48\sqrt{\pi}PS^{3}\sqrt{\frac{1}{CP}}+24\pi^{2}\sqrt{6}Q^{2}S}}\end{split}$

(43)

$\begin{split}&\Phi=\left(\frac{\partial M}{\partial Q}\right)_{S,P,C,J}\\\
=&\frac{\sqrt[4]{3}Q\left(\pi
C\left(3\sqrt{2\pi}Q^{2}\sqrt{\frac{1}{CP}}+8\sqrt{3}S\right)+2\sqrt{3}S^{2}\right)}{2\
2^{3/4}CS^{1/2}\sqrt{3\pi^{5/2}\sqrt{\frac{1}{CP}}\left(32J^{2}PS+3Q^{4}\right)+\frac{96\pi^{3/2}\left(4\pi^{2}J^{2}+S^{2}\right)}{\sqrt{\frac{1}{CP}}}+\frac{2\sqrt{6}S^{2}\left(4PS^{2}+3\pi
Q^{2}\right)}{C}+48\sqrt{\pi}PS^{3}\sqrt{\frac{1}{CP}}+24\pi^{2}\sqrt{6}Q^{2}S}}\end{split}$

(44)

$\begin{split}&V_{C}=\left(\frac{\partial M}{\partial P}\right)_{S,Q,J,C}\\\
=&\frac{C\left(\frac{1}{CP}\right)^{3/2}\left(96\pi^{3/2}CP\left(4\pi^{2}J^{2}+S^{2}\right)+16\sqrt{6}P^{2}S^{4}\sqrt{\frac{1}{CP}}+\pi^{5/2}\left(96J^{2}PS-9Q^{4}\right)+48\sqrt{\pi}PS^{3}\right)}{16\sqrt[4]{2}3^{3/4}\pi
S^{1/2}\sqrt{3\pi^{5/2}\sqrt{\frac{1}{CP}}\left(32J^{2}PS+3Q^{4}\right)+\frac{96\pi^{3/2}\left(4\pi^{2}J^{2}+S^{2}\right)}{\sqrt{\frac{1}{CP}}}+\frac{2\sqrt{6}S^{2}\left(4PS^{2}+3\pi
Q^{2}\right)}{C}+48\sqrt{\pi}PS^{3}\sqrt{\frac{1}{CP}}+24\pi^{2}\sqrt{6}Q^{2}S}}\end{split}$

(45)

$\begin{split}&\mu=\left(\frac{\partial M}{\partial C}\right)_{S,Q,P,J}\\\
=&-\frac{P\left(\frac{1}{CP}\right)^{3/2}\left(-96\pi^{3/2}CP\left(4\pi^{2}J^{2}+S^{2}\right)+16\sqrt{6}P^{2}S^{4}\sqrt{\frac{1}{CP}}+12\pi\sqrt{6}PQ^{2}S^{2}\sqrt{\frac{1}{CP}}+\pi^{5/2}\left(96J^{2}PS+9Q^{4}\right)+48\sqrt{\pi}PS^{3}\right)}{16\sqrt[4]{2}3^{3/4}\pi
S^{1/2}\sqrt{3\pi^{5/2}\sqrt{\frac{1}{CP}}\left(32J^{2}PS+3Q^{4}\right)+\frac{96\pi^{3/2}\left(4\pi^{2}J^{2}+S^{2}\right)}{\sqrt{\frac{1}{CP}}}+\frac{2\sqrt{6}S^{2}\left(4PS^{2}+3\pi
Q^{2}\right)}{C}+48\sqrt{\pi}PS^{3}\sqrt{\frac{1}{CP}}+24\pi^{2}\sqrt{6}Q^{2}S}}\end{split}$

(46)

$\begin{split}&\Omega_{b}=\left(\frac{\partial M}{\partial
J}\right)_{S,Q,P,C}\\\ =&\frac{4\
2^{3/4}\sqrt[4]{3}\pi^{3/2}JP\sqrt{\frac{1}{CP}}(4\pi
C+S)}{S^{1/2}\sqrt{3\pi^{5/2}\sqrt{\frac{1}{CP}}\left(32J^{2}PS+3Q^{4}\right)+\frac{96\pi^{3/2}\left(4\pi^{2}J^{2}+S^{2}\right)}{\sqrt{\frac{1}{CP}}}+\frac{2\sqrt{6}S^{2}\left(4PS^{2}+3\pi
Q^{2}\right)}{C}+48\sqrt{\pi}PS^{3}\sqrt{\frac{1}{CP}}+24\pi^{2}\sqrt{6}Q^{2}S}}\end{split}$

(47)

The thermodynamics of Kerr-Ads black holes have been studied in depth in kk ;
ll ; mm

###  2.3 CFT Thermodynamics

In this section, let us now review the mass of the CFT thermodynamics for the
single-charged and rotating black hole. The metric is set as nn ; oo ; pp ; qq
-

$ds^{2}=\omega^{2}(-dt^{2}+l^{2}d\Omega^{2}_{2}),$ (48)

The two-sphere line element is given as $d\Omega^{2}_{2}$ and $\omega$ is an
arbitrary dimensionless conformal factor. In some papers, we have noticed that
they have taken the conformal factor $\omega=R\l$ where $R$ is the curvature
of the radius. But by taking into consideration paper pp  and allowing it to
be a general parameter, the parameter $\omega$ allows us to examine a
holographic first law keeping Newton’s constant fixed and a varying
cosmological constant$\Lambda$.
The boundary sphere’s spatial volume is given as

$\mathcal{V}=\Omega_{2}(\omega l)^{2}$ (49)

For this volume $\mathcal{V}$ which is given in the CFT context, an analogous
pressure $p$ is also there so a work term $-pd\mathcal{V}$ exists. By using
the holographic dictionary in the bulk mass, entropy, temperature, electric
potential, angular momentum, and charge along with their analogous are given
as j ; bb ; qq

$E=\frac{M}{\omega},\quad\mathcal{T}=\frac{T}{\omega},\quad\mathcal{S}=S,\quad\mathcal{J}=J,\quad\varphi=\frac{\Phi}{\omega
l},\quad\mathcal{Q}=Ql.$ (50)

By using (22),(39), (50) we get the CFT energy as

$E^{2}=\frac{64\pi^{4}C^{2}\mathcal{J}^{2}+16\pi^{2}C^{2}\mathcal{S}^{2}+16\pi^{3}C\mathcal{J}^{2}\mathcal{S}+8\pi^{3}C\mathcal{Q}^{2}\mathcal{S}+8\pi
C\mathcal{S}^{3}+2\pi^{2}\mathcal{Q}^{2}\mathcal{S}^{2}+\pi^{4}\mathcal{Q}^{4}+\mathcal{S}^{4}}{4\pi^{2}C\mathcal{S}\mathcal{V}}$
(51)

The thermodynamic first law and the Smarr relation are related as

$dE=\mathcal{T}d\mathcal{S}+\varphi d\mathcal{Q}-\ pd\mathcal{V}+\mu dC+\Omega
d\mathcal{J}$ (52) $E=\mathcal{T}\mathcal{S}+\varphi\mathcal{Q}+\mu
C+\Omega\mathcal{J}$ (53)

Now, calculating the related quantities we can write them as

$\mathcal{T}=\left(\frac{\partial
E}{\partial\mathcal{S}}\right)_{\mathcal{Q},\mathcal{V},C}=\frac{16\pi^{2}C^{2}\left(\mathcal{S}^{2}-4\pi^{2}\mathcal{J}^{2}\right)+16\pi
C\mathcal{S}^{3}+2\pi^{2}\mathcal{Q}^{2}\mathcal{S}^{2}-\pi^{4}\mathcal{Q}^{4}+3\mathcal{S}^{4}}{4\pi\mathcal{S}\sqrt{C\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(2\pi^{2}\mathcal{J}^{2}+\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$

(54)

$\varphi=\left(\frac{\partial
E}{\partial\mathcal{Q}}\right)_{\mathcal{S},\mathcal{V},C}=\frac{\pi\mathcal{Q}\left(4\pi
C\mathcal{S}+\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)}{\sqrt{C\mathcal{S}\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(\pi^{2}\left(2\mathcal{J}^{2}+\mathcal{Q}^{2}\right)+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$

(55)

$-p=\left(\frac{\partial
E}{\partial\mathcal{V}}\right)_{\mathcal{S},\mathcal{Q},C}=-\frac{\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(\pi^{2}\left(2\mathcal{J}^{2}+\mathcal{Q}^{2}\right)+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}{4\pi\mathcal{V}^{3/2}\sqrt{CS}}$

(56)

$\mu=\left(\frac{\partial E}{\partial
C}\right)_{\mathcal{S},\mathcal{Q},\mathcal{V}}=\frac{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)-\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}{4\pi
C^{3/2}\sqrt{\mathcal{S}\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(\pi^{2}\left(2\mathcal{J}^{2}+\mathcal{Q}^{2}\right)+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$

(57)

$\Omega=\left(\frac{\partial
E}{\partial\mathcal{J}}\right)_{\mathcal{S},\mathcal{Q},\mathcal{V}}=\frac{8\pi^{2}\mathcal{J}(4\pi
C+\mathcal{S})}{C^{-1/2}\sqrt{\mathcal{S}\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(\pi^{2}\left(2\mathcal{J}^{2}+\mathcal{Q}^{2}\right)+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$

(58)

Here (56) is the CFT equation of state. The conjugate for the central charge
is the chemical potential $\mu$ (57)
From the bulk mass formula (29), we see that the internal energy formula (51)
can be formulated. Also, we can relate that the bulk thermodynamics first law
(30) corresponds to the conformal field theory (CFT) first law (52), where we
note that the appearance of the variable Newton’s constant is not there on
incorporating a variable cosmological constant $\Lambda$

##  3 List of thermodynamic ensembles in the CFT

In recent papers dd , the CFT states phase transitions that relate to the
charged AdS black hole were examined. Here we try to add one more variable
namely $\mathcal{J}$ and its conjugate $\Omega$ for the rotation to make it a
charge and rotating black holes which we will only look into the corresponding
CFT states of the KN -AdS black holes. The list of all the ensembles are-
fixed $(\mathcal{Q},\mathcal{J},\mathcal{V},C)\,$, fixed
$(\mathcal{Q},\mathcal{J},\mathcal{V},\mu)$, fixed
$(\mathcal{Q},\mathcal{J},p,C)$, fixed $(\mathcal{Q},\mathcal{J},p,\mu)$,
fixed $(\mathcal{Q},\Omega,\mathcal{V},C)$, fixed
$(\mathcal{Q},\Omega,\mathcal{V},\mu)$, fixed $(\mathcal{Q},\Omega,p,C)$,
fixed $(\mathcal{Q},\Omega,p,\mu)$, fixed
$(\varphi,\mathcal{J},\mathcal{V},C)$, fixed
$(\varphi,\mathcal{J},\mathcal{V},\mu)$, fixed $(\varphi,\mathcal{J},p,C)$,
fixed $(\varphi,\mathcal{J},p,\mu)$, fixed $(\varphi,\Omega,\mathcal{V},C)$,
fixed $(\varphi,\Omega,\mathcal{V},\mu$, fixed $(\varphi,\Omega,p,C)$, fixed
$(\varphi,\Omega,p,\mu)$ ensembles. Even though there are 16 ensembles to
examine, it is quite challenging to study all the ensembles so we only took 10
ensembles. and examined their phase transitions and criticality. The ensembles
are- fixed
$(\mathcal{Q},\mathcal{J},\mathcal{V},C)$,fixed$(\mathcal{Q},\Omega,\mathcal{V},C)$,
fixed $(\Phi,\Omega,\mathcal{V},C)$, fixed
$(\varphi,\mathcal{J},\mathcal{V},C)$, fixed $(\mathcal{Q},\mathcal{J},p,C)$,
fixed $(\mathcal{Q},\Omega,p,C)$, fixed $\varphi,\mathcal{J},p,C$, fixed
$(\varphi,\Omega,p,C)$, fixed $(\mathcal{Q},\mathcal{J},p,\mu)$, fixed
$(\varphi,\Omega,\mathcal{V},\mu)$ ensembles respectively.

###  3.1 Ensemble at fixed $(\mathcal{Q},\mathcal{J},\mathcal{V},C)$

In this canonical ensemble, we look into fixing the electric charge
$\mathcal{Q}$, the CFT volume $\mathcal{V}$, the angular momentum
$\mathcal{J}$, and the central charge $C$. The Helmholtz free energy is given
as

$F=E-\mathcal{T}\mathcal{S}=\frac{16\pi^{2}C^{2}\left(12\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+16\pi^{3}C\mathcal{S}\left(2J^{2}+\mathcal{Q}^{2}\right)+2\pi^{2}\mathcal{Q}^{2}\mathcal{S}^{2}+3\pi^{4}\mathcal{Q}^{4}-\mathcal{S}^{4}}{4\pi\sqrt{C\mathcal{S}\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(2\pi^{2}\mathcal{J}^{2}+\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$

(59)

and the temperature in this ensemble is given as:-

$\mathcal{T}=\frac{16\pi^{2}C^{2}\left(\mathcal{S}^{2}-4\pi^{2}\mathcal{J}^{2}\right)+16\pi
C\mathcal{S}^{3}+2\pi^{2}\mathcal{Q}^{2}\mathcal{S}^{2}-\pi^{4}\mathcal{Q}^{4}+3\mathcal{S}^{4}}{4\pi\mathcal{S}^{3/2}\sqrt{C\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(2\pi^{2}\mathcal{J}^{2}+\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$

(60)

From the CFT first law (52), the differentiation of $F$ satisfies-

$dF=dE-\mathcal{T}d\mathcal{S}-\mathcal{S}d\mathcal{T}=-\mathcal{S}d\mathcal{T}+\varphi
d\mathcal{Q}-pd\mathcal{V}+\mu dC+\Omega d\mathcal{J}$ (61)

From the above equation (61) we can see that $F$ is fixed at
$(\mathcal{T},\mathcal{Q},\mathcal{V},C,\mathcal{J})$.
We can now infer how the free energy $F$ behaves as a function of T for
different fixed values of $(\mathcal{Q},\mathcal{V},C,\mathcal{J})$. For this,
we can plot $F(T)$ parametrically using the entropy $\mathcal{S}$ as the
parameter for the fixed values. In the figure we plot for different values of
$\mathcal{Q}$ when keeping both $C$ and $\mathcal{J}$ fixed and also for
different values of $\mathcal{J}$ keeping both $\mathcal{Q}$, $C$ as constant
and lastly for different values of $C$ and keeping $\mathcal{Q}$ and
$\mathcal{J}$ as constant.

(a) We plot for the various values of $\mathcal{J}$ for the fixed
$\mathcal{Q}$, $\mathcal{V}$ and $C$. For $\mathcal{J}<\mathcal{J}_{crit}$
(blue, orange) we can see a first-order phase transition. For
$\mathcal{J}=\mathcal{J}_{crit}$ (green) there we see a second order phase
transition and for $\mathcal{J}>\mathcal{J}_{Crit}$(red), there are no phase
transitions.

(b) We plot for the various values of $\mathcal{Q}$ for fixed values of
$\mathcal{J}$, $\mathcal{V}$ and $C$. For
$\mathcal{Q}<\mathcal{Q}_{crit}$(blue, orange) we see a first-order phase
transition. For $\mathcal{Q}=\mathcal{Q}_{crit}$ we see a second order phase
transition and for $\mathcal{Q}>\mathcal{Q}_{crit}$, there are no phase
transitions

(c) We plot for the various values of $C$ for the fixed values of
$\mathcal{Q}$, $\mathcal{J}$, $\mathcal{V}$ and $C$. For $C>C_{crit}$ we see a
first-order phase transition. For $C=C_{crit}$ we see a second-order phase
transition and for $C<C_{crit}$ we see no phase transition.

Figure 1: Free energy $F$ vs. temperature $\mathcal{T}$ plot for the fixed
$(\mathcal{Q},\mathcal{J},\mathcal{V},C)$ ensemble in $d=4$.

We can see that in the three cases of plots displaying the swallow-tail
behavior. But one thing is noticed that this occurs only in
$\mathcal{Q}<\mathcal{Q}_{crit}$, $\mathcal{J}<\mathcal{J}_{crit}$ and for the
central charge case, its $C>C_{crit}$. the former two cases is seen in Plots
(1(a)) and (1(b)) and the later case is seen in Plot (1(c)) of Figure 1.
On the plot (1(a)) we have shown the free energy function concerning the
temperature $\mathcal{T}$ for various values of $\mathcal{J}$. Here we see
$\mathcal{J}<\mathcal{J}_{crit}$(blue, orange),
$\mathcal{J}=\mathcal{J}_{crit}$ (green) and
$\mathcal{J}>\mathcal{J}_{crit}$(red) with keeping some variables namely
$\mathcal{Q}$, $C$ and $\mathcal{V}$ as constant. We can see that the free
energy plot (1(a)) , we see "swallow-tail" shape for
$\mathcal{J}<\mathcal{J}_{crit}$, a kink like structure for
$\mathcal{J}=\mathcal{J}_{crit}$ and a smooth curve for
$\mathcal{J}>\mathcal{J}_{crit}$.For all the curves on the plot (1(a)) when
$T=0$, as $\mathcal{T}$ increases, the value of entropy $\mathcal{S}$ also
increases. On the curve where the swallow tail is seen, we can see the low
entropy state only near $\mathcal{T}=0$ and hence initially the lowest free
energy. Until it reaches the self-intersection point, it will continue to have
the lowest free energy. After crossing this point, for large entropy
$\mathcal{S}$ black holes with high entropy per degree of freedom, which are
lying along the vertical branch, will have the lowest free energy dominating
the canonical ensemble. For the value of $\mathcal{J}<\mathcal{J}_{crit}$ we
kind of see a first-order phase transition between the high and low entropy
states at the intersection temperature. As the value of the $\mathcal{J}$
increases, the swallow tail part diminishes until
$\mathcal{J}=\mathcal{J}_{crit}$ where the remains are only a kink-like
structure. The plot at the critical point
$(\mathcal{J}_{crit},\mathcal{T}_{crit}$, we see a second order phase
transition which can be seen to be dependant of $C$.

For Plot (1(b)) we see the same kind of pattern as Plot (1(a)). We see the
free energy plot as a function for temperature for different values of charge
namely $\mathcal{Q}<\mathcal{Q}_{crit}$ (blue, orange),
$\mathcal{Q}=\mathcal{Q}_{crit}$(green) and
$\mathcal{Q}>\mathcal{Q}_{crit}$(red) while we keep $C,\mathcal{V}$ and
$\mathcal{J}$ fixed. Same like the plot (1(a)), we see a swallow-tail like
behaviour for $\mathcal{Q}<\mathcal{Q}_{crit}$, then a structure like kink for
$\mathcal{Q}=\mathcal{Q}_{crit}$ and then a monotonous smooth curve for the
range $\mathcal{Q}>\mathcal{Q}_{crit}$. For $\mathcal{T}=0$, the entropy
increases as the temperature $\mathcal{T}$ increases. From the entropy formula
if we consider the small entropy $\mathcal{S}$ which is analogous to the CFT
states with small $\mathcal{S}/C$. Considering the swallowtail curve, only
near $\mathcal{T}=0$ we see a low-entropy state and having the minimum free
energy $F$. It will have low free energy as the temperature increases till
when it reaches the intersection point. After crossing this point, for the
large $\mathcal{S}$ black holes that are seen to be vertical which are the
analogous CFT state having higher entropy per degree of freedom, it becomes
the lowest free energy state. At the self-intersection point, a first-order
phase transition is seen between the considered low entropy and high entropy
states for each considered value of $\mathcal{Q}<\mathcal{Q}_{crit}$. As the
value of $\mathcal{Q}$ further increases, we see that the swallowtail shrinks
till the $\mathcal{Q}$ reaches the $\mathcal{Q}_{crit}$ point, and now we see
just a kink like structure in the plot.For $\mathcal{Q}>\mathcal{Q}_{crit}$ we
see a smooth curve.

We have also considered changes in central charge $C$ which can be seen in
plot (1(c)) while keeping $\mathcal{Q}$, $\mathcal{J}$ and $\mathcal{V}$
fixed. We note three different perspective namely $C<C_{crit}$(red),
$C=C_{crit}$ (green) and $C>C_{crit}$(orange, blue). The plot (1(c)) kind of
shows the same kind of structure just like plot (1(a)) and (1(b)) but a bit
dissimilar as they show a first-order phase transition for the low- to high
entropy degrees of freedom for $C>C_{crit}$, then a second order phase
transition for $C=C_{crit}$ and last we see a single phase for $C<C_{crit}$ As
we decrease the value of the central charge $C$ the swallow tail behavior will
keep on decreasing until $C$ reaches a critical value after which we don’t see
a swallow tail but only a kink. If we compare all three plots then we see that
the first order phase transition that is the formation of swallow tail occurs
at $\mathcal{J}<\mathcal{J}_{crit}$ for plot (1(a)) and
$\mathcal{Q}<\mathcal{Q}_{crit}$ for plot (1(b)) and
$\mathcal{C}>\mathcal{C}_{crit}$ for plot (1(c)).

###  3.2 Ensemble at fixed $(\mathcal{Q},\Omega,\mathcal{V},C)$

In this grand canonical ensemble, we try to fix the angular velocity $\Omega$,
the electric charge $\mathcal{Q}$, the CFT volume $\mathcal{V}$, and the
central charge $C$. The free energy here is given as

$\begin{split}&G=E-\mathcal{T}\mathcal{S}-\Omega\mathcal{J}\\\
&=\frac{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+16\pi^{3}C\mathcal{Q}^{2}\mathcal{S}+2\pi^{2}\mathcal{Q}^{2}\mathcal{S}^{2}+3\pi^{4}\mathcal{Q}^{4}-\mathcal{S}^{4}}{4\pi\sqrt{C\mathcal{S}\mathcal{V}}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(2\pi^{2}\mathcal{J}^{2}+\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}\end{split}$
(62)

and the temperature is given as

$\mathcal{T}=\frac{16\pi^{2}C^{2}\left(\mathcal{S}^{2}-4\pi^{2}\mathcal{J}^{2}\right)+16\pi
C\mathcal{S}^{3}+2\pi^{2}\mathcal{Q}^{2}\mathcal{S}^{2}-\pi^{4}\mathcal{Q}^{4}+3\mathcal{S}^{4}}{4\pi\sqrt{C\mathcal{V}}\mathcal{S}^{3/2}\sqrt{16\pi^{2}C^{2}\left(4\pi^{2}\mathcal{J}^{2}+\mathcal{S}^{2}\right)+8\pi
C\mathcal{S}\left(2\pi^{2}\mathcal{J}^{2}+\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)+\left(\pi^{2}\mathcal{Q}^{2}+\mathcal{S}^{2}\right)^{2}}}$
(63)

Comparing from the CFT first law (52), and differentiating $G$ from (62) we
get

$dG=dE-\mathcal{T}d\mathcal{S}-\mathcal{S}d\mathcal{T}-\Omega
d\mathcal{J}-\mathcal{J}d\Omega=-\mathcal{S}d\mathcal{T}+\varphi
d\mathcal{Q}-pd\mathcal{V}+\mu dC-\mathcal{J}d\Omega$ (64)

From the above equation (64) we can see that $G$ is fixed at
$(\mathcal{T},\mathcal{Q},\mathcal{V},C,\Omega)$ We can see that the free
energy $G$ behaves as a function of the temperature $\mathcal{T}$ for the
fixed values of $(\mathcal{Q},\Omega,\mathcal{V},C)$. For this ensemble, we
can plot $G(\mathcal{T})$ parametrically by using the help of the entropy
$\mathcal{S}$.In the figure below we plot different values of $C$ that is
central charge, while we keep both the angular velocity $\Omega$ and charge
$\mathcal{Q}$ fixed and also for various values of $\mathcal{Q}$ while keeping
$C$ and $\Omega$ fixed and lastly for various values of $\Omega$ hence while
keeping $Q$ and $C$ constant. A total of three cases is seen in Figure 2 for
the plots displaying a "swallow-tail" behavior. We notice here that this
swallow tail kind of behavior happens only for cases like
$\mathcal{Q}<\mathcal{Q}_{crit}$, $\Omega<\Omega_{crit}$ and for the central
charge its a bit different namely $C>C_{crit}$. The first two cases implying
with $\mathcal{Q}$ and $\Omega$ is seen in plot (2(a)) and (2(b)) and the one
with the change in $C$ is seen in plot (2(c)).

(a) We plot for the various values of $\Omega$ for the fixed $\mathcal{Q}$,
$\mathcal{V}$ and $C$. For $\Omega<\Omega_{crit}$ (blue, orange) we can see a
first-order phase transition. For $\Omega=\Omega_{crit}$ (green), we see a
second-order phase transition, and for $\Omega>\Omega_{crit}$(red), there are
no phase transitions.

(b) We plot for the various values of $\mathcal{Q}$ for the fixed $\Omega$,
$\mathcal{V}$ and $C$. For $\mathcal{Q}<\mathcal{Q}_{crit}$ (blue, orange) we
can see a first-order phase transition. For $\mathcal{Q}=\mathcal{Q}_{crit}$
(green) there we see a second order phase transition and for
$\mathcal{Q}>\mathcal{Q}_{crit}$(red), there are no phase transitions.

(c) We plot for the various values of $C$ for the fixed $\mathcal{Q}$,
$\mathcal{V}$ and $\Omega$. For $C>C_{crit}$ (blue, orange) we can see a
first-order phase transition. For $C=C_{crit}$ (green), we see a second-order
phase transition and for $C<C_{Crit}$(red), there are no phase transitions.

Figure 2: Free energy $G$ vs. temperature $\mathcal{T}$ plot for the fixed
$(\mathcal{Q},\Omega,\mathcal{V},C)$ ensemble in $d=4$

On the first plot, we have a plot (2(a)), the free energy function for the
temperature for various values of $\Omega$. Here we see the picture as for
$\Omega<\Omega_{crit}$ (blue, orange) for $\omega=\omega_{crit}$(green) and
for $\Omega>\Omega_{crit}$ (red) while all for keeping $\mathcal{Q},C$ and
$\mathcal{V}$ fixed. In this plot (2(a)) we see that the swallow tail is seen
for $\Omega<\Omega_{crit}$, a kink-like figure for $\Omega=\Omega_{crit}$ and
at last a smooth monotonous curve for $\Omega>\Omega_{crit}$. When
$\mathcal{T}=0$ for all curves on plot (2(a)), as the temperature increases,
the value of $\mathcal{S}$ also increases. we can see a low entropy stage,
where the swallow tail is seen nearby $\mathcal{T}=0$. when it reaches the
self-intersection point, it continues to have the lowest free energy. For
large entropy $\mathcal{S}$ black holes are seen after crossing the self-
intersection point, lying along the vertical branch, having the lowest free
energy in the ensemble. For this range that is $\Omega<\Omega_{crit}$ we see
the first order phase transition. As the value of $\omega$ increases, the
swallow tail part is seen starting to vanish thereby remaining only a kink-
like figure at $\Omega=\Omega_{crit}$. The plot at the critical point that is
$(\Omega_{crit},\mathcal{T}_{crit})$ we see a second-order phase transition.
For Plot (2(b)) we kind of see the same kind of pattern as seen in plot
(2(a)). Here also we plot the free energy for temperature for different values
of $\mathcal{Q}$.

For the second plot (2(b)), we kind of notice the same kind of pattern as seen
in Plot (2(a)). Here we try to plot the free energy as a function of
temperature for various values of charges namely
$\mathcal{Q}<\mathcal{Q}_{crit}$ (blue, orange),
$\mathcal{Q}=\mathcal{Q}_{crit}$ (green) and $\mathcal{Q}>\mathcal{Q}_{crit}$
(red) while we keep central charge $C$, CFT volume $\mathcal{V}$ and angular
velocity $\Omega$ as constant or fixed. We see a "swallow-tail" behaviour for
$\mathcal{Q}<\mathcal{Q}_{crit}$, a kink like structure for
$\mathcal{Q}=\mathcal{Q}_{cirt}$ and then a smooth monotonous curve for
$\mathcal{Q}>\mathcal{Q}_{crit}$. Starting from $\mathcal{T}=0$ for all the
curves, the entropy $\mathcal{S}$ values increases along the curve as the
value of $T$ increases. First, if we consider the swallowtail curve, only at
near $\mathcal{T}=0$ we see a low entropy state and also having the minimum
free energy $G$. It will continue to have low free energy as the temperature
increases till it reaches the self-intersection point. After we cross this
self-intersection point, the curves that are seen to be vertical for large
$\mathcal{S}$ black holes which seem to be conjugate for the CFT states having
higher entropy per degree of freedom, become the lowest state for free energy.
We see a first-order phase transition at the self-intersection point between
the low and high entropy for various values in the range
$\mathcal{Q}<\mathcal{Q}_{crit}$. As the value of $\mathcal{Q}$ increases we
can see that the swallow tail decreases till it reaches a kink for
$\mathcal{Q}=\mathcal{Q}_{crit}$ and then for $\mathcal{Q}>\mathcal{Q}_{crit}$
we see a smooth curve.
We also have considered the changes in central charges $C$ which can be seen
in plot (2(c)) while we keep $\mathcal{Q}$, $\Omega$ and $\mathcal{V}$ as
fixed. We kind of note three different regimes namely $C<C_{crit}$ (red),
$C=C_{crit}$ (green) and $C>C_{crit}$ (orange, blue). The third plot (2(c))
shows kind of same structure as plot (2(a)) and plot (2(b)) but a bit
dissimilar is seen as they kind of show first-order phase transition for the
low to high entropy degrees of freedom for the central charges in the ranges
$C>C_{crit}$. We see a second order phase transition when $C=C_{crit}$ and
last we see a single phase or we can say a smooth monotonous curve for
$C<C_{crit}$. As we decrease the value of the central charge $C$ we notice
that the "swallow-tail" behavior keeps on decreasing until $C$ reaches a
critical kind of value which looks like a "kink" at the critical value after
this we do not see any "swallow-tail" kind of structure.
If we kind of compare the three plots we see that the phase transition of
first order occurs at $\Omega<\Omega_{crit}$ for plot (2(a)),
$\mathcal{Q}<\mathcal{Q}_{crit}$ for plot (2(b)) and $C>C_{crit}$ for plot
(2(c)).

###  3.3 Ensemble at fixed $(\varphi,\Omega,\mathcal{V},C)$

In this ensemble we kind of fix the angular velocity $\Omega$, the electric
potential $\Phi$, the CFT volume $\mathcal{V}$, and the central charge $C$. We
can derive the free energy as

$\begin{split}&W=E-\mathcal{T}\mathcal{S}-\varphi\mathcal{Q}-\Omega\mathcal{J}\\\
&=-\frac{2\sqrt{\mathcal{S}}\Omega\sqrt{\frac{C\mathcal{S}^{2}\varphi^{2}\mathcal{V}^{2}\Omega^{2}}{16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}}}\left(\frac{4\pi
C\varphi^{2}\mathcal{V}(4\pi
C+\mathcal{S})}{16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}}+4\pi
C+\mathcal{S}\right)}{8\pi^{3/2}C\varphi\sqrt{\mathcal{V}}\sqrt{4\pi
C+\mathcal{S}}}-\frac{16\pi\sqrt{C}\sqrt{\mathcal{S}}\varphi^{2}\sqrt{\mathcal{V}}\sqrt{4\pi
C+\mathcal{S}}}{8\pi^{3/2}\sqrt{16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}}}+\frac{1}{\sqrt{\pi}}\sqrt{\frac{\mathcal{S}(4\pi
C+\mathcal{S})^{3}\left(4\pi
C\left(\varphi^{2}\mathcal{V}+4\pi\right)+\mathcal{S}\left(4\pi-\mathcal{V}\Omega^{2}\right)\right)^{2}}{C\mathcal{V}\left(16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{3}}}\\\
&+\frac{\sqrt{\frac{\mathcal{S}(4\pi C+\mathcal{S})^{3}\left(4\pi
C\left(\varphi^{2}\mathcal{V}+4\pi\right)+\mathcal{S}\left(4\pi-\mathcal{V}\Omega^{2}\right)\right)^{2}}{C\mathcal{V}\left(16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{3}}}\left(4\pi
C\mathcal{S}^{2}\left(\mathcal{V}^{2}\Omega^{2}\left(\varphi^{2}+2\Omega^{2}\right)-4\pi\mathcal{V}\left(\varphi^{2}+9\Omega^{2}\right)+112\pi^{2}\right)+3\mathcal{S}^{3}\left(\mathcal{V}\Omega^{2}-4\pi\right)^{2}\right)}{8\pi^{3/2}(4\pi
C+\mathcal{S})^{2}\left(4\pi
C\left(\varphi^{2}\mathcal{V}+4\pi\right)+\mathcal{S}\left(4\pi-\mathcal{V}\Omega^{2}\right)\right)}-\\\
&\frac{\sqrt{\frac{\mathcal{S}(4\pi C+\mathcal{S})^{3}\left(4\pi
C\left(\varphi^{2}\mathcal{V}+4\pi\right)+\mathcal{S}\left(4\pi-\mathcal{V}\Omega^{2}\right)\right)^{2}}{C\mathcal{V}\left(16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{3}}}\left(64\pi^{3}C^{2}\mathcal{S}\left(-2\varphi^{2}\mathcal{V}-3\mathcal{V}\Omega^{2}+20\pi\right)+256\pi^{4}C^{3}\left(4\pi-\varphi^{2}\mathcal{V}\right)\right)}{8\pi^{3/2}(4\pi
C+\mathcal{S})^{2}\left(4\pi
C\left(\varphi^{2}\mathcal{V}+4\pi\right)+\mathcal{S}\left(4\pi-\mathcal{V}\Omega^{2}\right)\right)}\end{split}$

(65)

and the temperature in this ensemble is given as

$\begin{split}&\mathcal{T}=-\frac{\sqrt{\frac{\mathcal{S}(4\pi
C+\mathcal{S})^{3}\left(4\pi
C\varphi^{2}\mathcal{V}+16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{2}}{C\mathcal{V}\left(16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{3}}}\left(-128\pi^{3}C^{2}\mathcal{S}\varphi^{2}\mathcal{V}-256\pi^{4}C^{3}\varphi^{2}\mathcal{V}+8\pi
C\mathcal{S}^{2}\mathcal{V}^{2}\Omega^{4}+144\pi^{2}C\mathcal{S}^{2}\mathcal{V}\Omega^{2}+3\mathcal{S}^{3}\mathcal{V}^{2}\Omega^{4}-24\pi\mathcal{S}^{3}\mathcal{V}\Omega^{2}\right)}{8\pi^{3/2}\mathcal{S}(4\pi
C+\mathcal{S})^{2}\left(4\pi
C\varphi^{2}\mathcal{V}+16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)}\\\
&\frac{\left(-192\pi^{3}C^{2}\mathcal{S}\mathcal{V}\Omega^{2}+1280\pi^{4}C^{2}\mathcal{S}+1024\pi^{5}C^{3}+448\pi^{3}C\mathcal{S}^{2}+48\pi^{2}\mathcal{S}^{3}\right)\sqrt{\frac{\mathcal{S}(4\pi
C+\mathcal{S})^{3}\left(4\pi
C\varphi^{2}\mathcal{V}+16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{2}}{C\mathcal{V}\left(16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{3}}}}{8\pi^{3/2}\mathcal{S}(4\pi
C+\mathcal{S})^{2}\left(4\pi
C\varphi^{2}\mathcal{V}+16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)}\\\
&-\frac{\sqrt{\frac{\mathcal{S}(4\pi C+\mathcal{S})^{3}\left(4\pi
C\varphi^{2}\mathcal{V}+16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{2}}{C\mathcal{V}\left(16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)^{3}}}\left(4\pi
C\mathcal{S}^{2}\varphi^{2}\mathcal{V}^{2}\Omega^{2}+16\pi^{2}C\mathcal{S}^{2}\varphi^{2}\mathcal{V}\right)}{8\pi^{3/2}\mathcal{S}(4\pi
C+\mathcal{S})^{2}\left(4\pi
C\varphi^{2}\mathcal{V}+16\pi^{2}C-\mathcal{S}\mathcal{V}\Omega^{2}+4\pi\mathcal{S}\right)}\end{split}$

(66)

If we see the CFT first law (52), differentiating the free energy expression W
(65), we get

$dW=dE-\mathcal{T}d\mathcal{S}-\mathcal{S}d\mathcal{T}-\Omega
d\mathcal{J}-\mathcal{J}d\Omega-\varphi
d\mathcal{Q}-\mathcal{Q}d\varphi=-\mathcal{S}d\mathcal{T}-\mathcal{Q}d\varphi-
pd\mathcal{V}+\mu dC-\mathcal{J}d\Omega$ (67)

From the above equation (67) we can notice that the $W$ is kind of fixed at
$(\mathcal{T},\varphi,\mathcal{V},C,\Omega)$. We see from the above equation
(67) that the free energy behaves as a function of temperature $\mathcal{T}$
for the fixed values of $(\varphi,\Omega,\mathcal{V},C)$. For this ensemble
given here, we can plot parametrically $W(\mathcal{T})$ for the entropy
$\mathcal{S}$.

(a) We plot for various values of $\varphi$ keeping $C$, $\Omega$,
$\mathcal{V}$ as constant. For $\varphi<\varphi_{crit}$ the free energy
consists of two branches. At $W=0$, there is a first-order phase transition.
For $\varphi\geq\varphi_{crit}$, the curve lies below $W=0$ and no phase
transition takes place.

(b) We plot for the various vales of $\Omega$ while keeping $\varphi$, $C$ and
$\mathcal{V}$ constant.For $\Omega<\Omega_{crit}$, the free energy consists of
two branches. At $W=0$, there is a first-order phase transition. For
$\Omega\geq\Omega_{crit}$ there is a single branch that lies above $W=0$ and
has no phase transition.

(c) We plot for the various values of $\mathcal{V}$ keeping $\varphi$,
$\Omega$, $C$. For $\mathcal{V}<\mathcal{V}$, the free energy consists of two
branches. At $W=0$, there is a first-order phase transition. For
$\mathcal{V}\geq\mathcal{V}_{crit}$ there is a single branch that lies above
$W=0$ and has no phase transition.

Figure 3: Free energy $W$ vs. temperature $\mathcal{T}$ plot for the fixed
($\varphi$,$\Omega$,$\mathcal{V}$,C) ensemble in $d=4$ dimensions

From the expression of the free energy and temperature, we can plot the free
energy $W(\mathcal{T})$ using the entropy $\mathcal{S}$ parametrically for the
fixed value of $(\Omega,\varphi,\mathcal{V},C)$ as we can see in the Figure 3.
In the plot (3(a)) we can see that the $W-\mathcal{T}$ displays a certain
different kind of behavior above and below a certain critical value
$\varphi_{crit}$. We see for the value of $\varphi\geq\varphi_{crit}$ (green,
red), we can see that the free energy is a single-valued function as a
function of temperature, where $W\leq 0$ and the curve cuts the $W$ axis. also
for the value of $\varphi<\varphi_{crit}$ (orange, blue), we can see that the
free energy curve consists of an upper or lower branch that meets at a cusp
which actually corresponds to the low entropy (small black holes) and the high
entropy states (large black holes) respectively.
We see that the blue and the orange curve change sign for the free energy at
$W=0$ which corresponds to a first-order phase transition. When the free
energy $W<0$ then the large - entropy which is termed as "deconfined" seems to
be dominating. But when $W>0$, we see that the "confined state is more
preferred." We note that the confinement/deconfinement phase transition is
analogous to the generalized Hawking-Page phase transition between the AdS
space containing thermal radiation and the large AdS black holes. We see that
the Hawking-Page transition also exists for charged and rotating black holes
which was originally discovered for AdS-Schwarzschild black holes.
We see the plot (3(b)) of figure 3 We can see the variation of the free energy
$W$ with respect to $\mathcal{T}$ with the help of entropy $\mathcal{S}$
parametrically. We can note that for various values of $\Omega$, there is
either a single-valued behavior of the free energy or a bifurcation point from
where two sets of branches emerge. We can calculate the turning point or the
cusp as

$\left(\frac{\partial\mathcal{T}}{\partial
S}\right)_{\varphi,\Omega,\mathcal{V},C}=0$ (68)

which in turn gives the corresponding entropy and also the temperature of the
turning point as( $S_{cusp}$, $\mathcal{T}(S_{cusp})$). From this equation, we
get one of the solutions for $S_{cusp}$ as

$S_{cusp}=-\frac{4\pi\left(C\mathcal{V}\varphi^{2}+4\pi
C\right)}{4\pi-\Omega^{2}\mathcal{V}}$ (69)

Therefore putting the value of $S_{cusp}$ (69) in the temperature (66), we
get:-

$\begin{split}&\mathcal{T}=\frac{\left(192\pi^{3}\left(C\mathcal{V}\varphi^{2}+4C\pi\right)^{3}\left(V\Omega^{2}-4\pi\right)^{2}\right)\sqrt{-\frac{\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)^{2}\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}{CV\left(4\pi-V\Omega^{2}\right)\left(\frac{4\pi
V\left(CV\varphi^{2}+4C\pi\right)\Omega^{2}}{4\pi-V\Omega^{2}}+16C\pi^{2}-\frac{16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}}}{\left(4\pi-V\Omega^{2}\right)^{2}\left(16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{2}\right)}\\\
&-\frac{\left(64C\pi^{3}\left(CV\varphi^{2}+4C\pi\right)^{2}\left(V^{2}\left(\varphi^{2}+2\Omega^{2}\right)\Omega^{2}+112\pi^{2}\right)\right)\sqrt{-\frac{\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)^{2}\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}{CV\left(4\pi-V\Omega^{2}\right)\left(\frac{4\pi
V\left(CV\varphi^{2}+4C\pi\right)\Omega^{2}}{4\pi-V\Omega^{2}}+16C\pi^{2}-\frac{16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}}}{\left(4\pi-V\Omega^{2}\right)\left(16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{2}\right)}\\\
&-\frac{\left(256C^{2}\pi^{4}\left(CV\varphi^{2}+4C\pi\right)\left(-2V\varphi^{2}-3V\Omega^{2}+20\pi\right)\right)\left(256C^{3}\pi^{4}\left(4\pi-V\varphi^{2}\right)\right)\sqrt{-\frac{\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)^{2}\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}{CV\left(4\pi-V\Omega^{2}\right)\left(\frac{4\pi
V\left(CV\varphi^{2}+4C\pi\right)\Omega^{2}}{4\pi-V\Omega^{2}}+16C\pi^{2}-\frac{16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}}}{\left(16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{2}\right)}\\\
&\frac{\left(4\pi-V\Omega^{2}\right)\sqrt{-\frac{\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)^{2}\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}{CV\left(4\pi-V\Omega^{2}\right)\left(\frac{4\pi
V\left(CV\varphi^{2}+4C\pi\right)\Omega^{2}}{4\pi-V\Omega^{2}}+16C\pi^{2}-\frac{16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{3}}}\left(4\pi
V\left(\varphi^{2}+9\Omega^{2}\right)\right)}{16\pi^{2}\left(CV\varphi^{2}+4C\pi\right)\left(4C\pi\left(V\varphi^{2}+4\pi\right)-4\pi\left(CV\varphi^{2}+4C\pi\right)\right)\left(4C\pi-\frac{4\pi\left(CV\varphi^{2}+4C\pi\right)}{4\pi-V\Omega^{2}}\right)^{2}}\end{split}$

(70)

How ever if we just simplify the equation (70), it reduces to zero. So from
the $T(S_{cusp})$ we can extract that

$4\pi-\mathcal{V}\Omega\leq 0\quad
i.e\quad\Omega_{c}=\frac{2\sqrt{\pi}}{\sqrt{\mathcal{V}}}$ (71)

Hence if the above condition satisfies then we can see a single-valued Gibbs
free energy is obtained.

From the plot (3(c)) we also notice that there is a point where the free
energy changes its sign. This does indicate phase transition where we know
that when a system has a lower free energy it is a more thermodynamically
preferred region. So we kind of say that when the temperature increases from
zero of the CFT, in this ensemble the confined state is dominated until it
reaches a turning point where the free energy $W$ changes its sign from a
positive value to a negative value. There only the system is seen to have a
first-order phase transition and after this, the system seems to be dominated
by the de-confined state large entropy. This is kind of similar to the
Hawking-Page transition.

###  3.4 Ensemble at fixed $(\varphi,\mathcal{J},\mathcal{V},C)$

In this ensemble we try to fix the electric potential $\varphi$, the angular
velocity $\mathcal{J}$, CFT volume $\mathcal{V}$, and the central charge $C$.
We can write the free energy as

$X=E-\mathcal{T}\mathcal{S}-\varphi\mathcal{Q}$ (72)

We will not be able to write down explicitly the free energy and temperature
expressions as they are quite lengthy and complicated. From the CFT first law,
we notice that from the free energy expression (52), we put the value in the
result when differentiating (72), we get:-

$dX=dE-\mathcal{T}d\mathcal{S}-\mathcal{S}d\mathcal{T}-\varphi
d\mathcal{Q}-\mathcal{Q}d\varphi=-\mathcal{S}d\mathcal{T}-\mathcal{Q}d\varphi-
pd\mathcal{V}+\mu dC$ (73)

We see that the free energy $X$ (73) is a kind of fixed at
$(\mathcal{T},\varphi,\mathcal{V},C,\Omega)$. We can note that the free energy
$X$ behaves as a function of the temperature $\mathcal{T}$ for the various
fixed values of $(\varphi,\mathcal{J},\mathcal{V},C)$. For this system, we can
plot parametrically $X(\mathcal{T})$ by using the entropy $\mathcal{S}$. In
the figure 5 we plot different values of $\mathcal{J}$ while keeping electric
potential $\varphi$ and the central charge $C$ fixed.Also, we plot for various
values of $C$ where we keep both $\varphi$ and $\mathcal{J}$ fixed. Also, we
plot the graphs seen in Figure 5(a) for various values of $\varphi$ and keep
$C$ and $\mathcal{J}$ fixed. We see a total of three cases is there for the
plots showing the swallowtail behavior. But from the plots, we can notice one
thing, this criticality occurs in $\mathcal{J}<\mathcal{J}_{crit}$,
$\varphi<\varphi_{crit}$ for the angular momentum and electric potential case
and for the central charge it occurs at $C>C_{crit}$.

(a) We plot for the various values of $\mathcal{J}$ for the fixed $\varphi$,
$\mathcal{V}$ and $C$. For $\mathcal{J}<\mathcal{J}_{crit}$ (blue, orange) we
can see a first-order phase transition. For $\mathcal{J}=\mathcal{J}_{crit}$
(green) there we see a second order phase transition and for
$\mathcal{J}>\mathcal{J}_{crit}$(red), there are no phase transitions.

(b) We plot for the various values of $C$ for the fixed $\varphi$,
$\mathcal{V}$ and $\mathcal{J}$. For $C<C_{crit}$ (blue, orange) we can see a
first-order phase transition. For $C=C_{crit}$ (green) we see a second-order
phase transition and for $C>C_{crit}$(red), there are no phase transitions.

Figure 4: Free energy F vs temperature $\mathcal{T}$ plot for fixed
($\varphi$,$\mathcal{J}$,$\mathcal{V}$,C) ensemble in $d=4$

For the first two cases which implies $\mathcal{J}$ and $C$, we see it in plot
(4(a)) and (4(b)) and the the change in $\varphi$, we see it in plot (5(a))
On the first plot (4(a)), we see that the free energy is plotted for the
temperature by using the entropy $\mathcal{S}$ parametrically for the various
values of $\mathcal{J}$. Here we note that for the values where
$\mathcal{J}<\mathcal{J}_{crit}$ (blue,orange), then for
$\mathcal{J}=\mathcal{J}_{crit}$ (green) and lastly for
$\mathcal{J}>\mathcal{J}_{crit}$ (red) while we keep $\varphi$, $C$ and
$\mathcal{V}$ as constant. Here for the plot (4(a)), we see that a swallow-
tail is generated for various values of $\mathcal{J}<\mathcal{J}_{crit}$, then
a kink-like structure for $\mathcal{J}=\mathcal{J}_{crit}$ and then a smooth
monotonous curve for various values of $\mathcal{J}>\mathcal{J}_{crit}$. We
see that for all the curves when $\mathcal{T}=0$, when the temperature
increases, the value for the entropy also increases. we can see at a low
entropy range, it continues to have the lowest free energy when it reaches the
self-intersection point. For the black holes having large entropy
$\mathcal{S}$, it is noticed that the black hole after crossing the self-
intersection point, the points lying along the vertical branch is seen to have
the lowest free energy in the considered ensemble. For this range that is we
see when $J<J_{crit}$, we kind of see a first-order phase transition. Now as
the value of $\mathcal{J}$ increases, the swallow tail part of the curve is
seen to vanish thereby now only remaining a kink-like structure at
$\mathcal{J}=\mathcal{J}_{crit}$. The plot at the critical point
$(\mathcal{J}_{crit},\mathcal{T}_{crit})$ is actually the second-order phase
transition. For the plot (4(b)), we see the same kind of pattern as seen in
the plot (4(a)). But there is a catch that we talk about below. We also plot
the free energy $X$ with respect to the temperature for various values of $C$.
In the plot (4(b)) we note that we have considered the changes in central
charges while we have kept the $\mathcal{J}$, $\varphi$ and $\mathcal{V}$ as
fixed or we can say as constant. Here we seem to have three regions namely the
regions where $C<C_{crit}$ (red), $C=C_{crit}$ (green), and $C>C_{crit}$
(orange, blue). This plot seems a bit different from the plot(4(a)) in that a
dissimilarity occurs as they show the first-order phase transitions when the
value of the central charges $C>C_{crit}$. Then we also see a second-order
phase transition when the value of the central charge is $C=C_{crit}$ and at
last the red curve which is the smooth monotonous curve is due to the central
charge when the value is $C<C_{crit}$. As we decrease the value of the central
charge, we notice that the swallow-tail behavior tends to decrease until $C$
reaches a kind of critical value which looks like a kink at the critical value
after which we do not see any kind of swallow tail kind of structure.

(a) We plot for the various values of $\varphi$ for the fixed $\mathcal{J}$,
$\mathcal{V}$ and $C$. Here for all the plots $\varphi<\varphi_{crit}$

(b) We plot for the various values of $\varphi$ for the fixed $C$,
$\mathcal{V}$ and $\mathcal{J}$. Blue plot corresponds to
$\varphi<\varphi_{crit}$. Orange plot corresponds to $\varphi=\varphi_{crit}$
and green plot corresponds to $\varphi>\varphi_{crit}$

Figure 5: Free energy F vs temperature $\mathcal{T}$ plot for fixed
($\varphi$,$\mathcal{J}$,$\mathcal{V}$,C) ensemble in $d=4$

For plot (5(a)) and (5(b)), we plot the free energy for the temperature
parametrically with entropy for various values of $\mathcal{J}$, $C$,
$\varphi$ and $\mathcal{V}$. We see for plot (5(a)), by keeping $\mathcal{J}$,
$C$ and $\mathcal{V}$ constant, we plot for various values of $\varphi$. Here
in this plot, we notice three phases for all the values of $\varphi$. For plot
(5(b)), we plot for similar values of $\mathcal{J}$, $C$ and $\mathcal{V}$,
but different values of $\varphi$. Here we get three plots where for the blue
plot ($\varphi<\varphi_{crit}$) we get three phases and then for the orange
and green plot ($\varphi\geq\varphi_{crit}$), we get only one phase.

##  4 Other ensembles

In the previous sections we have discussed certain ensembles and their free
energy namely for fixed $(\mathcal{Q},\mathcal{J},\mathcal{V},C)$, fixed
$(\mathcal{Q},\Omega,\mathcal{V},C)$, fixed $(\varphi,\Omega,\mathcal{V},C)$
and lastly fixed $(\varphi,\mathcal{J},\mathcal{V},C)$ ensembles. Now in
addition to all these ensembles we also have taken 6 more ensembles, precisely
the p ensembles, and one ensemble taking all the dependent variables as seen
below from $F_{1}-F_{6}$ namely

$\begin{split}&\text{fixed}\quad(\mathcal{Q},\mathcal{J},p,C):\quad
F_{1}=E-\mathcal{T}\mathcal{S}+p\mathcal{V}=\varphi\mathcal{Q}+\Omega\mathcal{J}+\mu
C+p\mathcal{V}\\\ &\text{fixed}\quad(\mathcal{Q},\Omega,p,C):\quad
F_{2}=E-\mathcal{T}\mathcal{S}-\Omega\mathcal{J}-p\mathcal{V}=\varphi\mathcal{Q}+\mu
C+p\mathcal{V}\\\ &\text{fixed}\quad(\varphi,\mathcal{J},p,C):\quad
F_{3}=E-\mathcal{T}\mathcal{S}-\varphi\mathcal{Q}-p\mathcal{V}=\Omega\mathcal{J}+\mu
C+p\mathcal{V}\\\ &\text{fixed}\quad(\varphi,\Omega,p,C):\quad
F_{4}=E-\mathcal{T}\mathcal{S}-\varphi\mathcal{Q}-\Omega\mathcal{J}-p\mathcal{V}=\mu
C+p\mathcal{V}\\\ &\text{fixed}\quad(\mathcal{Q},\mathcal{J},p,\mu):\quad
F_{5}=E-\mathcal{T}\mathcal{S}-\mu
C+p\mathcal{V}=\varphi\mathcal{Q}+\Omega\mathcal{J}+p\mathcal{V}\\\
&\text{fixed}\quad(\varphi,\Omega,\mathcal{V},\mu):\quad
F_{6}=E-\mathcal{T}\mathcal{S}-\varphi\mathcal{Q}-\varphi\mathcal{J}-\mu
C=0\end{split}$ (74)

(a) Plot 1

(b) Plot 2

(c) Plot 3

(d) Plot 4

(e) Plot 5

Figure 6: These other ensembles are plotted for the temperature for $d=4$
dimensions corresponding to the ensembles at fixed
$(\mathcal{Q},\mathcal{J},p,C)$, fixed $(\mathcal{Q},\Omega,p,C)$, fixed
$(\varphi,\mathcal{J},p,C)$, fixed $(\varphi,\Omega,p,C)$, and for fixed
$(\mathcal{Q},\mathcal{J},p,\mu)$ respectively (left to right)

In the figure 6, we have seen that the free energy is being plotted against
the temperature for the first five ensembles for some constant parameter
values. The top part from left to right are the fixed
$(\mathcal{Q},\mathcal{J},p,C)$, $(\mathcal{Q},\Omega,p,C)$,
$(\varphi,\mathcal{J},p,C)$ and the bottom two from left to right are given as
$(\varphi,\Omega,p,C)$ and $(\mathcal{Q},\mathcal{J},p,\mu)$. We have not
plotted here for the $(\varphi,\Omega,\mathcal{V},\mu)$ ensemble because the
associated free energy $F_{6}$ is seen to be as zero that is $(F_{6}\equiv 0)$
due to the Euler equation expansion.
Certain things we can note from Figure 6 is that all the five ensembles do not
display any kind of critical phenomena or phase transitions. Therefore we can
say that we do not see any critical behavior in the ensembles taken or to be
precise the $p-\mathcal{V}$ criticality in the fixed p ensembles for the
thermal states or the CFT. Further in the three ensembles at fixed
$(\mathcal{Q},\mathcal{J},\mathcal{V},C)$,
$(\mathcal{Q},\Omega,\mathcal{V},C)$ and $(\mathcal{Q},\mathcal{J},p,\mu)$, we
observe two branches in the free energy plots: the lower branch corresponds to
a state with a low value of $S/C$ which associates to the small type of ADS
black holes and the upper branch corresponds to the high value of $S/C$ which
corresponds to the large type of AdS black holes. The lower branch has the
least free energy and hence this phase is dominating in this respective
ensembles.

##  5 Conclusion

In the beginning, we review very briefly the the mass-energy formulas for the
extended phase space thermodynamics, then for the mixed thermodynamics, and
lastly for the CFT thermodynamics for the Kerr-Newman AdS black hole
comprising both the charge and rotating element dd ; ii ; dd1 . Then we
investigated the various phase transitions for the dual holographic CFT states
for the Kerr-Newman AdS black hole taking in various ensembles. Note that we
were not able to take the entire ensemble as the interplay between the
$(\mathcal{J},\Omega)$ and $(\mathcal{Q},\varphi)$ is quite complex, so we
could only investigate a few ensembles. In the ensemble
$(\mathcal{Q},\mathcal{J},\mathcal{V},C)$ in section 3.1 we have plotted the
free energy $F$ for the temperature parametrically with the entropy
$\mathcal{S}$. We have plot three different plots seen in Figure 1 that we
plot for various values of electric charge $\mathcal{Q}$ when keeping the
$\mathcal{J}$, $\mathcal{V}$ and the central charge $C$ as constant. For this,
we have observed a first-order phase transition for the range of
$\mathcal{Q}<\mathcal{Q}_{crit}$. For when $\mathcal{Q}=\mathcal{Q}_{crit}$,
we observe a kink-like structure which is described as a second-order phase
transition. At last, for $\mathcal{Q}>\mathcal{Q}_{crit}$ we see a smooth
monotonous curve and there is no swallowtail or a kink anywhere. Now when we
plot various values of $\mathcal{J}$, we get a first-order phase transition
which is represented by a swallowtail for the range of angular momentum when
$\mathcal{J}<\mathcal{J}_{crit}$ and then a second order phase transition for
the value when $\mathcal{J}=\mathcal{J}_{crit}$ and it is visualized as a kink
like structure and then we observe a smooth curve for the value of
$\mathcal{J}$ when $\mathcal{J}>\mathcal{J}_{crit}$. Lastly, for this ensemble
we plot for the various values of the central charge $C$, keeping
$\mathcal{J}$, $\mathcal{Q}$, and $\mathcal{V}$ constant. When $C>C_{crit}$ we
get the swallow tail-like structure which is the first-order phase transition
at the point when the vertical line cuts the horizontal line. For
$C=C_{crit}$, we get the second order phase transition which is seen as a
kink, and then a smooth monotonous curve for the central charge range that is
$C<C_{crit}$
Now for the ensemble $(\mathcal{Q},\Omega,\mathcal{V},C)$ in section 3.2, we
also have plotted the $G(\mathcal{T})$ graph for various values and have
obtained three plots seen in Figure 2. Firstly we have plotted for various
values of $\Omega$, keeping $\mathcal{Q}$, $\mathcal{V}$ and $C$ as constant.
For this then we get a swallowtail-like structure when the value of $\Omega$
is $\Omega<\Omega_{crit}$. This is also called the first-order phase
transition. Now when $\Omega=\Omega_{crit}$ we get a kink-like structure that
corresponds to the second-order phase transition and then for
$\Omega>\Omega_{crit}$ we get a smooth monotonous curve. Now when we plot for
the various values of $\mathcal{Q}$ with various values as constant namely
$\Omega$, $\mathcal{V}$ and $C$, we see a first-order phase transition for
when $\mathcal{Q}<\mathcal{Q}_{crit}$ which is seen as a swallow tail
structure. Then when $\mathcal{Q}=\mathcal{Q}_{crit}$ we see a kink-like
structure which is second order phase transition and then when
$\mathcal{Q}>\mathcal{Q}_{crit}$ we get a smooth curve like structure. Now for
this ensemble if we plot the curve for various values of the central charge
$C$ for constant values of $\mathcal{Q}$, $\Omega$, and $\mathcal{V}$. We get
the first order phase transition for $C>C_{crit}$ which we see a swallowtail
behavior. Then for $C=C_{crit}$, we get the second-order phase transition
which also visually looks like a kink. and then lastly for $C<C_{crit}$ we get
a smooth monotonous curve.
In the ensemble $(\varphi,\Omega,\mathcal{V},C)$ of section 3.3, we have
plotted the $W(\mathcal{T})$ plot parametrically with entropy. When we plot
the various values of $\varphi$ while keeping the other parameters $\Omega$,
$\mathcal{V}$ and $C$ as constant. We notice that for
$\varphi>\varphi_{crit}$, the free energy plot is single-valued where $W\leq
0$ and the curve cuts the $W$ axis. Now for the value of
$\varphi<\varphi_{crit}$, we notice that the free energy curve consists of an
upper and lower branch. For this branch, it changes sign at $W=0$ which is the
first order phase transition. When $W<0$ it is said to be deconfined and when
$W>0$, it is said to be confined. Now when we plot for various values of
$\omega$, there is a single value behavior of the Gibbs free energy and then
there we see a bifurcation point where two branches appear. When the
temperature increases from zero of the CFT, the ensemble in the confined state
is dominated until we get a turning point where the free energy changes sign.
Here only the first-order phase transition takes place. We can see the same
kind of analogy when we plot for various values of $\mathcal{V}$. We also note
that the (de)confinement phase transition does not depend on the central
charge as we get no point where we get a single phase so the phase transition
means that $\Omega$ and $\varphi$ depend on the CFT volume.

Now, section 3.4, in the ensemble $(\varphi,\mathcal{J},\mathcal{V},C)$, we
have parametrically plot the $X(\mathcal{T})$ curve with the help of the
entropy $\mathcal{S}$. Now when we have plotted for various values of
$\mathcal{J}$, while having certain parameters constant like $\varphi$,
$\mathcal{V}$, $C$. Here for $\mathcal{J}<\mathcal{J}_{crit}$ we see a swallow
tail-like structure which is the first order phase transition. Then we see a
second-order phase transition which looks like a kink and is at the value
where $\mathcal{J}=\mathcal{J}_{crit}$. Then when the value of
$\mathcal{J}>\mathcal{J}_{crit}$ we see a smooth curve-like structure. Now
then for various values of $C$ keeping the $\varphi$, $\mathcal{V}$ and $C$ as
constant. Here we have seen a first-order phase transition that is a
swallowtail-like structure when the value of the central charge is
$C>C{crit}$. Then a second-order phase transition at the kink when
$C=C_{crit}$. and then a smooth monotonous curve when $C<C_{crit}$ where we
see no phase transition. Now when we plot for various values of $\varphi$, we
get the threee phases for all the values of $\varphi$. When
$\varphi<\varphi_{crit}$, we see three phases and when
$\varphi\geq\varphi_{crit}$, we get only one phase.
We also plot for the various $p$ ensembles in section 4. For six ensembles we
examine the phase transition but we only get for five since the free energy
for the fixed ($\varphi,\Omega,\mathcal{V},\mu$) ensemble gives zero
$(F_{6}=0)$. So, for the remaining five ensembles, the plot didn’t show any
kind of criticality or phase transitions. So there is no $p-\mathcal{V}$
criticality for the fixed $p$ ensembles for the thermal states or the CFT. The
CFT phase transition of charged and rotating black holes matches to some
extend to the charged black hole and rotating black hole. This type of
behaviour shows that there should be some underlying universality in the phase
transitions by using the AdS/CFT formalism.

In future endeavors, one can also examine the charge and rotating black holes
in a more generalized form where we can also include the $\mu$ containing
ensembles that were not included in this paper. One can also study more in-
depth for each ensemble and check their coexistence plots, and specific heat.
We can also extend the study to a different formalism, namely the RPST
(restricted phase space thermodynamics). We can also examine thermodynamic
topology by studying the phase transitions of various ensembles taken in this
paper.

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