Observing Black Hole Phase Transitions in Extended Phase Space and Holographic Thermodynamics Approaches from Optical Features

The action for Einstein-Maxwell gravity in $(n+1)$-dimensional AdS spacetime is given by

$\displaystyle S=\frac{1}{16\pi G_{N}}\int d^{n+1}x\sqrt{-g}\left[R-\mathcal{F}^{2}+\frac{n(n-1)}{L^{2}}\right],$

(6)

where $F$ is the $U(1)$ gauge field strength tensor and $\Lambda=-\frac{n(n-1)}{2L^{2}}$ is the negative cosmological constant with the length scale of the AdS space $L$. A spherical symmetric BH solution from this action is

$\displaystyle ds^{2}$

$\displaystyle=$

$\displaystyle-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\omega^{2}_{n-1},$

(7)

where

$\displaystyle f(r)$

$\displaystyle=$

$\displaystyle 1+\frac{r^{2}}{L^{2}}-\frac{m}{r^{n-2}}+\frac{q^{2}}{r^{2n-4}}.$

(8)

Note that $d\omega^{2}_{n-1}$ is the metric of unit $n-1$ sphere. The parameter $m$ is related to the mass $M$ of BH as follows

$\displaystyle M=\frac{(n-1)\omega_{n-1}}{16\pi G_{N}}m,$

(9)

where $\omega_{n-1}$ is the volume of the unit $n-1$ sphere. The electric charge of BH $Q$ can be written in the form of the parameter $q$ as

$\displaystyle Q=\frac{(n-1)\omega_{n-1}}{8\pi G_{N}}\eta q,$

(10)

where $\eta$ is given by

$\displaystyle\eta=\sqrt{\frac{2(n-2)}{n-1}}.$

(11)

For spherical symmetric and static charged BH, one can choose the gauge potential as

$\displaystyle\mathcal{A}=\left(-\frac{1}{\eta}\frac{q}{r^{n-2}}+\Phi\right)dt.$

(12)

Here, $\mathcal{A}$ is fixed and set to vanish at the horizon surface $r_{+}$, so that the electric potential $\Phi$ becomes

$\displaystyle\Phi=\frac{1}{\eta}\frac{q}{r_{+}^{n-2}}.$

(13)

The Hawking temperature of BH and its corresponding entropy are associated with its geometrical quantities as

	$\displaystyle T$	$\displaystyle=$	$\displaystyle\frac{\kappa}{2\pi}=\frac{n-2}{4\pi r_{+}}\left(1+\frac{n}{n-2}\frac{r_{+}^{2}}{L^{2}}-\frac{q^{2}}{r_{+}^{2n-4}}\right),$		(14)
	$\displaystyle S$	$\displaystyle=$	$\displaystyle\frac{A}{4G_{N}}=\frac{\omega_{n-1}r_{+}^{n-1}}{4G_{N}},$		(15)

where $\kappa$ and $A$ denote the surface gravity and horizon area, respectively.

Recall that the partition function in the grand canonical ensemble $\mathcal{Z}$ is the Laplace transform of the density of states $g(E,N)$ as follows

$\displaystyle\mathcal{Z}=\int g(E,N)e^{-\beta(E-\mu N)}dEdN,$

(16)

where $\beta,E,\mu$ and $N$ denote the inverse temperature, internal energy, chemical potential and number of particles in the system, respectively. To obtain the density of states, we apply an inverse Laplace transform to $\mathcal{Z}$ and then use the steepest descent method. Thus, we have

$\displaystyle g(E,N)\sim\mathcal{Z}e^{\beta(E-\mu N)}.$

(17)

As the thermal entropy $S$ defined as the logarithm of the number of states, the above equation can give the grand potential, $\Omega\equiv-T\ln\mathcal{Z}$, in the form of thermodynamic variables as

$\displaystyle\Omega=E-TS-\mu N,$

(18)

The first law of thermodynamics satisfies

$\displaystyle d\Omega=-SdT-pd\mathcal{V}-Nd\mu.$

(19)

Thus, we treat $T,\mathcal{V}$ and $\mu$ as independent thermodynamic variables so that we can express the grand potential as the function of these three variables, i.e., $\Omega=\Omega(T,\mathcal{V},\mu)$. Applying the Legendre transformation to obtain the Helmholtz free energy of the canonical ensemble, $F(T,\mathcal{V},N)=\Omega(T,\mathcal{V},\mu)+\mu N$, we have

$\displaystyle F=E-TS.$

(20)

With the first law of thermodynamics, the infinitesimal change in $F$ reads

$\displaystyle dF=-SdT-pd\mathcal{V}-\mu dN.$

(21)

In gravitational physics, the partition function of BHs can be approximated from the on-shell Euclidean action $S_{E}$ as follows [69]

$\displaystyle\mathcal{Z}\sim e^{-S_{E}}.$

(22)

To evaluate the Einstein-Maxwell action in Eq. (6) with the potential $A_{t}$ fixed at infinity, the Gibbons-Hawking boundary term vanishes due to the strength tensor $\mathcal{F}$ becoming zero. Consequently, the partition function $\mathcal{Z}$ can be simply calculated from the bulk action without any additional term. Using the subtraction method, the resulting thermodynamic potential is in the form [70, 71, 72]

$\displaystyle-T\ln\mathcal{Z}\equiv\Omega=M-TS-\Phi Q.$

(23)

Comparing Eqs. (18) with (23), one may identify $E$, $\mu$ and $N$ of the thermal systems with $M$, $\Phi$ and $Q$ of the charged BH. It is important to note that the thermodynamic potential corresponding to the grand potential $\Omega$, in the above equation, is derived by calculating the on-shell action with potential $A_{t}$ fixed at the boundary of spacetime. Recall that the potential $\Phi$ is given by taking $r\to\infty$ into Eq. (12). Alternatively, the thermodynamic potential corresponding to the Helmholtz free energy $F$ can be obtained by fixing the charge $Q$ at the spacetime boundary instead. In this approach, the Gibbons-Hawking term no longer vanishes but becomes significant and contributes to the on-shell action $S_{E}$. The resulting Euclidean action calculation with fixed $Q$ at the boundary yields the Helmholtz free energy $F$ as the corresponding thermodynamic potential:

$\displaystyle-T\ln\mathcal{Z}\equiv F=M-TS.$

(24)

As discussed in the previous subsection, it is indeed possible to associate the thermodynamic variables of BH with the geometric properties of its spacetime. However, the Euler’s theorem for a homogeneous function suggests that the Smarr formula in Eq. (2) for BHs in the spacetime with non-zero $\Lambda$ become inconsistent with the first law of black hole thermodynamics unless variations in $\Lambda$ are taken into account. Using Eqs. (2) and (3) and allowing for variations in the cosmological constant $\Lambda$, we obtain the Smarr formula and the variation of mass in the form

	$\displaystyle M$	$\displaystyle=$	$\displaystyle\frac{n-1}{n-2}TS+\Phi Q-\frac{2}{n-2}PV,$		(25)
	$\displaystyle dM$	$\displaystyle=$	$\displaystyle TdS+\Phi dQ+VdP,$		(26)

respectively. Note that the computations concerning the Smarr formula and its associated first law derived from the scaling argument are detailed in Appendix A of the manuscript.

As presented in Eq. (26), the BH mass $M$ is a function of $S,Q$ and $P$, thus it should be interpreted as the enthalpy $H(S,Q,P)$ rather than the internal energy $E(S,Q,V)$. Using the Legendre transformation $E=H-PV$, treating $M$ to be enthalpy leads to the first law of black hole thermodynamics in the extended phase space of the form

$\displaystyle dE=TdS+\Phi dQ-PdV.$

(27)

Significantly, the first law can now describe the change in internal energy during processes associated with changes in volume within a black hole’s event horizon when it absorbs energy or emits Hawking radiation through the $PdV$ term, akin to conventional thermodynamics.

Let us consider black hole thermodynamics in the canonical ensemble within the extended phase space approach. Eliminating $L$ in Eq. (14) by identifying $\displaystyle P=\frac{3}{8\pi G_{N}L^{2}}$, we obtain the equation of state for a $4$-dimensional charged AdS-BH as follows [18]

$\displaystyle P=\frac{T}{2r_{+}}-\frac{1}{8\pi r_{+}^{2}}+\frac{q^{2}}{8\pi r_{+}^{4}}.$

(28)

By comparing the above equation with the equation of state for vdW fluids, one can find that $q$ relates to the gas’s constant in the vdW equation of state [18]. Using Eq. (28), we plot the isotherm curves for different temperatures, as shown in the left panel in Fig. 1. This depiction presents the critical behavior in the $P-r_{+}$ plane. Note that the critical values $r_{c},T_{c}$ and $P_{c}$ of this system satisfy the following conditions:

$\displaystyle\frac{\partial P}{\partial r_{+}}=0,\ \ \ \text{and}\ \ \ \frac{\partial^{2}P}{\partial r_{+}^{2}}=0.$

(29)

Solving these two conditions, we obtain

$\displaystyle r_{c}=\sqrt{6}q,\ \ \ T_{c}=\frac{\sqrt{6}}{18\pi q},\ \ \ P_{c}=\frac{1}{96\pi q^{2}}.$

(30)

Since $P$ should have positive values for $r_{+}>0$, there exists a particular value of temperature $T_{0}$ which is the lower bound of temperature where $P=0$ at a particular value of horizon radius $r_{0}$. Namely,

$\displaystyle T_{0}=\frac{\sqrt{3}}{18\pi q},\ \ \ r_{0}=\sqrt{3}q.$

(31)

At $T$ below $T_{c}$, the isotherm curves in the $P-r_{+}$ plane show the appearance of a local minimum $P_{\text{min}}$ and a local maximum $P_{\text{max}}$ at the horizon radii $r_{\text{min}}$ and $r_{\text{max}}$, respectively. Using Eq. (30) and fixing $q=1$, the critical parameters are given by $r_{c}=2.45$, $T_{c}=0.0433$, and $P_{c}=0.0033$. To illustrate the characteristics of the system, we present the curves for $T$ less than, equal to, and greater than $T_{c}$. As shown in Fig. 1, the curves for $T=0.0310$ (below the critical point), $T=T_{c}$, and $T=0.0500$ (above the critical point) are depicted. For $T=0.0310$, we find that $P_{\text{min}}=0.0001$ and $P_{\text{max}}=0.0016$, corresponding to $r_{\text{min}}=1.7391$ and $r_{\text{max}}=4.6615$, respectively.

According to the Le Chatelier principle [73, 74, 75, 76], the response function associated to the isotherm curves in $P-r_{+}$ plane is the isothermal compressibility $\kappa_{T}$, which is defined as

$\displaystyle\kappa_{T}=-\frac{1}{V}\frac{\partial V}{\partial P}\Big{|}_{T}=\frac{12\pi r_{+}^{4}}{2\pi Tr_{+}^{3}-r_{+}^{2}+2q^{2}},$

(32)

where $V=\frac{4}{3}\pi r_{+}^{3}$ is thermodynamic volume. The right panel in Fig. 1 displays $r_{+}$ dependence of $\kappa_{T}$ corresponding to isotherm curves in $P-r_{+}$ plane in the left panel. The results reveal three branches of BHs below $T_{c}$: Small BH (red solid curve), Intermediate BH (green solid curve) and Large BH (blue solid curve) correspond to the ranges of $r_{+}<r_{\text{min}},r_{\text{min}}<r_{+}<r_{\text{max}}$ and $r_{+}>r_{\text{max}}$, respectively. Here $r_{\text{min}}$ and $r_{\text{max}}$ is the event horizon radii that $\kappa_{T}$ diverge. Note that these values of $r_{+}$ can be obtained by setting the denominator in Eq. (32) to be zero, namely $2\pi Tr_{+}^{3}-r_{+}^{2}+2q^{2}=0$, and finding its roots. The Small BH and Large BH (both have $\kappa_{T}>0$) are mechanically stable against microscopic fluctuation while the Intermediate BH ($\kappa_{T}<0$) is unstable. The authors in [77] have shown that the unstable Intermediate BH in the $P-V$ plane can be replaced by a straight line at $P=P_{f}$ obeying the Maxwell equal area law, where the Small-Large BHs first-order phase transition occurs. Note that $P_{f}$ can be called as the Hawking-Page pressure. Defining the reduced variables as

$\displaystyle p=\frac{P}{P_{c}},\ \ \ t=\frac{T}{T_{c}}\ \ \ \text{and}\ \ \ v=\frac{V}{V_{c}},$

(33)

an analytic expression for Hawking-Page pressure is given by [77]

$\displaystyle p^{*}=\left[1-2\cos\left(\frac{\arccos\left(1-t^{2}\right)+\pi}{3}\right)\right]^{2},$

(34)

where $p^{*}=P_{f}/P_{c}$ denotes the reduced Hawking-Page pressure. Considering the isotherm curve with $T=0.0310$, for example, which is less than $T_{c}$ as shown in Fig. 1, the corresponding reduced temperature $t=0.7159$ can be put into the above equation such that the reduced pressure at which the first-order phase transition occurs is at $p^{*}=0.4384$, corresponding to $P_{f}=0.0014$. Moreover, the horizon radii for Small and Large BHs at the first-order phase transition are respectively expressed as follows

	$\displaystyle\frac{r_{S}}{r_{c}}$	$\displaystyle=$	$\displaystyle\frac{2}{t}\cos^{2}\phi-\sqrt{\frac{4}{t^{2}}\cos^{4}\phi-\frac{\sqrt{2}}{t}\cos\phi},$		(35)
	$\displaystyle\frac{r_{L}}{r_{c}}$	$\displaystyle=$	$\displaystyle\frac{2}{t}\cos^{2}\phi+\sqrt{\frac{4}{t^{2}}\cos^{4}\phi-\frac{\sqrt{2}}{t}\cos\phi},$		(36)

where $\phi=\frac{\pi-\theta}{3}$ and $\cos{\theta}=\frac{\sqrt{2}}{2}t$.

To investigate the global stability for charged AdS-BH associated with the isotherm curves in the $P-r_{+}$ plane, one needs to consider the free energy as a function of bulk pressure $P$ with the temperature $T$ held fixed. In the fixed charge ensemble, treating $M$ as the internal energy results in the thermodynamic potential being the Helmholtz free energy $F$, as is typical in standard phase space. However, in this section, where $M$ is treated as the enthalpy, the thermodynamic potential is instead the Gibbs free energy $G$. We express $G$ in term of $r_{+}$ with fixed $T$ in the following form

	$\displaystyle-T\ln\mathcal{Z}\equiv G$	$\displaystyle=$	$\displaystyle E+PV-TS$		(37)
		$\displaystyle=$	$\displaystyle M-TS$
		$\displaystyle=$	$\displaystyle\frac{2q^{2}-\pi r_{+}^{3}T+r_{+}^{2}}{3r_{+}}.$

By using $dM$ from Eq. (26) with the charge fixed, the infinitesimal change of the Gibbs free energy is

$\displaystyle dG=dM-TdS-SdT,$

(38)

can be changed to be in the form

$\displaystyle dG=-SdT+VdP.$

(39)

This relation suggests that $G=G(T,P)$ leading to the expression for entropy and thermodynamic volume given by

$\displaystyle\left(\frac{\partial G}{\partial T}\right)_{P}=-S,\ \ \ \left(\frac{\partial G}{\partial P}\right)_{T}=V.$

(40)

We plot $G$ against $P$ for a fixed $T$ in Fig. 2. It is important to note that the point $r_{+}$ where $\kappa_{T}$ becomes divergent corresponds to a cusp in the $G(P)$ graph. This feature reflects the second-order phase transition resulting from $\kappa_{T}=-\frac{1}{V}\left(\frac{\partial^{2}G}{\partial P^{2}}\right)_{T}$. As shown in the left panel in Fig. 2, the free energy in the case of $T<T_{c}$, exhibits the swallowtail behavior with two cusps at $P_{\text{min}}$ and $P_{\text{max}}$ corresponding to the horizon radii that $\kappa_{T}$ diverges $r_{\text{min}}$ and $r_{\text{max}}$, respectively. Considering the bulk pressure $P$ running from a low value where only Large-BH exists, Small and Intermediate BHs both emerge at $P=P_{\text{min}}$ with free energy larger than the Large BH. Obviously, Large BH is thermodynamically preferred in the region of $P<P_{f}$. It turns out that the system encounters the first-order phase transition at $P_{f}$ such that Small BH phase becomes thermodynamically preferred when $P>P_{f}$. The middle panel in Fig. 2 shows the case of $T=T_{c}$ where the Small-Large BHs transition becomes the second-order type of phase transition at $P=P_{c}$. The right panel in Fig. 2 shows that only one phase of BH exists at high temperature $T>T_{c}$, here at $T=0.0500$, where $\kappa_{T}>0$ at all values of $P$. Remarkably, by identifying Small, Intermediate, and Large BHs as liquid, metastable, and gas phases, respectively, the behaviors of the BH phase transition is in a similar way as what occurs in the vdW fluid.

As mentioned in section I, Introduction, the pressure $P$ and thermodynamic volume $V$ characterizing BHs in the bulk through the extended phase space approach do not directly correspond to the pressure $p$ and volume $\mathcal{V}$ of the dual field theory on the boundary within the AdS/CFT correspondence. Additionally, the Smarr formula, which relates the mechanical quantities of BHs to thermodynamic variables, depends on the number of spacetime dimensions, this feature is typically absent in the Euler equation for ordinary matter. Recently, the holographic thermodynamics approach has provided insights into effectively addressing these two issues in associating the black hole thermodynamic properties with its dual field theory counterpart. In this section, we examine the thermodynamics of CFT, which is holographically dual to the charged AdS-BH, within the holographic thermodynamics approach.

Introducing the central charge $\mathcal{C}$ and its chemical potential $\mu_{\mathcal{C}}$ as a new pair of thermodynamic variable in the large-$N$ gauge theory, the author in [31] suggests that the scaling behavior of its internal energy $E$ may be written as

$\displaystyle E(\alpha S,\alpha^{0}\mathcal{V},\alpha B_{i},\alpha\mathcal{C})=\alpha E(S,\mathcal{V},B_{i},\mathcal{C}),$

(41)

where $B_{i}$ denote the conserved quantities, such as charge and angular momentum. The Euler equation and its corresponding first law due to the Euler scaling argument can be obtained by using Eqs. (125) and (126) in Appendix A as

	$\displaystyle E$	$\displaystyle=$	$\displaystyle TS+\nu^{i}B_{i}+\mu_{\mathcal{C}}\mathcal{C},$		(42)
	$\displaystyle dE$	$\displaystyle=$	$\displaystyle TdS-pd\mathcal{V}+\nu^{i}dB_{i}+\mu_{\mathcal{C}}d\mathcal{C},$		(43)

where

$\displaystyle T=\left(\frac{\partial E}{\partial S}\right)_{\mathcal{V},B_{i},\mathcal{C}},p=-\left(\frac{\partial E}{\partial\mathcal{V}}\right)_{S,B_{i},\mathcal{C}},\nu^{i}=\left(\frac{\partial E}{\partial B_{i}}\right)_{S,\mathcal{V},\mathcal{C}},\mu_{\mathcal{C}}=\left(\frac{\partial E}{\partial\mathcal{C}}\right)_{S,\mathcal{V},B_{i}}.$

(44)

Note that the $p\mathcal{V}$ term is absent in the Euler equation but the $pd\mathcal{V}$ term appear in the first law. Since the variation of $E$ in Eq. (42) should be equal to the first law in Eq (43), i.e., $TdS+SdT+\nu^{i}dB_{i}+B_{i}d\nu^{i}+\mu_{\mathcal{C}}d\mathcal{C}+\mathcal{C}d\mu_{\mathcal{C}}=TdS-pd\mathcal{V}+\nu^{i}dB_{i}+\mu d\mathcal{C}$, this implies the Gibbs-Duhem equation as follows

$\displaystyle SdT+B_{i}d\nu^{i}+\mathcal{C}d\mu_{\mathcal{C}}=-pd\mathcal{V}.$

(45)

The grand potential is given by

	$\displaystyle\Omega$	$\displaystyle=$	$\displaystyle E-TS-\nu^{i}B_{i},$		(46)
		$\displaystyle=$	$\displaystyle(TS+\nu^{i}B_{i}+\mu_{\mathcal{C}}\mathcal{C})-TS-\nu^{i}B_{i},$
		$\displaystyle=$	$\displaystyle\mu_{\mathcal{C}}\mathcal{C},$

where we have substituted $E$ by using Eq. (42).

Since $\mathcal{C}$ depends on both $L$ and $G_{N}$ due to the holographic dictionary as shown in Eq. (5), thus varying $\mathcal{C}$ in the dual field theory should be equivalent to variation of $\Lambda$ and $G_{N}$ in the gravity side. By including $\Lambda$ and $G_{N}$ into the mass formula of AdS-BH in Eq. (127), an infinitesimal change of $M$ can be obtained from Eq. (126) as

$\displaystyle dM=\frac{\kappa}{8\pi G_{N}}dA+\Phi dQ+\frac{\Theta}{8\pi G_{N}}d\Lambda-(M-\Phi Q)\frac{dG_{N}}{G_{N}},$

(47)

where an additional partial derivative of $M$ with respect to $G_{N}$ is $\frac{\partial M}{\partial G_{N}}=-\frac{(M-\Phi Q)}{G_{N}}$. Substituting $\Theta$ in term of other variables by using the Smarr formula in Eq. (2) with $\Lambda$ expressed in term of $L$, the above equation becomes

	$\displaystyle dM=\frac{\kappa}{2\pi}d\left(\frac{A}{4G_{N}}\right)+\frac{\Phi}{L}d(QL)-\frac{M}{n-1}\frac{dL^{n-1}}{L^{n-1}}+\left(M-\frac{\kappa A}{8\pi G_{N}}-\Phi Q\right)\frac{d(L^{n-1}/G_{N})}{L^{n-1}/G_{N}}.$
			(48)

Thermodynamic variables of dual field theory can be identified with geometric quantities of AdS-BH as

$\displaystyle E=M,\ \ \ \tilde{\Phi}=\frac{\Phi}{L},\ \ \ \tilde{Q}=QL,\ \ \ \mathcal{V}\sim L^{n-1},\ \ \ \mathcal{C}\sim\frac{L^{n-1}}{G_{N}}.$

(49)

Substituting them into Eq. (48) and comparing with Eq. (43), we obtain

$\displaystyle dE=TdS+\tilde{\Phi}d\tilde{Q}-pd\mathcal{V}+\mu_{\mathcal{C}}d\mathcal{C},$

(50)

where

$\displaystyle p=\frac{E}{(n-1)\mathcal{V}},\ \ \ \mu_{\mathcal{C}}=\frac{1}{\mathcal{C}}\left(E-TS-\tilde{\Phi}\tilde{Q}\right).$

(51)

Notably, the former relation in (51) expresses the pressure $p$, in the field theory side, satisfies the equation of state in Eq. (4), while the latter gives the Euler equation corresponding to Eq.(42) with identifying $\nu=\tilde{\Phi}$ and $B=\tilde{Q}$. It is interesting to note that the CFT in the field theory side from the approach of holographic thermodynamics can live in the curved spacetime with the curvature radius $R$, which has a value not necessarily equal to the bulk AdS radius $L$. This can be seen by redefining the thermodynamic variables as [31, 32]

$\displaystyle E=M\frac{L}{R},\ \ \ T=\frac{\kappa}{2\pi}\frac{L}{R},\ \ \ S=\frac{A}{4G_{N}},\ \ \ \tilde{\Phi}=\frac{\Phi}{L}\frac{L}{R},\ \ \ \tilde{Q}=QL.$

(52)

From the resulting first law, we have $\mathcal{V}\sim R^{n-1}$ and $\displaystyle\mathcal{C}\sim\frac{L^{n-1}}{G_{N}}$. Thus, the volume $\mathcal{V}$ and central charge $\mathcal{C}$ of the dual CFT are now independent due to the choice of rescaling as shown in Eq. (52).

It is worthwhile to review here some interesting results about the thermodynamics of CFT that are holographically dual to charged AdS-BH within the novel holographic thermodynamics approach. For simplicity, we define the dimensionless quantities:

$\displaystyle x=\frac{r_{+}}{L},\ \ \ y=\frac{q}{L^{n-2}}.$

(53)

The thermodynamic quantities of the dual CFT, which is in $n$-dimensional spacetime of curvature radius $R$, can be written in parametric equations of $x$ and $y$ as follows [32]

	$\displaystyle E$	$\displaystyle=$	$\displaystyle\frac{(n-1)\mathcal{C}x^{n-2}}{R}\left(1+x^{2}+\frac{y^{2}}{x^{2n-4}}\right),$		(54)
	$\displaystyle T$	$\displaystyle=$	$\displaystyle\frac{n-2}{4\pi Rx}\left(1+\frac{n}{n-2}x^{2}-\frac{q^{2}}{x^{2n-4}},\right),$		(55)
	$\displaystyle S$	$\displaystyle=$	$\displaystyle 4\pi\mathcal{C}x^{n-1},$		(56)
	$\displaystyle\tilde{Q}$	$\displaystyle=$	$\displaystyle 2\eta(n-1)\mathcal{C}y,$		(57)
	$\displaystyle\tilde{\Phi}$	$\displaystyle=$	$\displaystyle\frac{1}{\eta R}\frac{y}{x^{n-2}},$		(58)
	$\displaystyle\mu_{\mathcal{C}}$	$\displaystyle=$	$\displaystyle\frac{x^{n-2}}{R}\left(1-x^{2}-\frac{y^{2}}{x^{2n-4}}\right).$		(59)

Here, the field theory under consideration lives in the spacetime with $n=3$ dimensions. The CFT is in the ensemble with fixed $(\tilde{Q},\mathcal{V},\mathcal{C})$, corresponding to the canonical ensemble. The temperature and heat capacity of the dual CFT are given by

	$\displaystyle T$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi Rx}\left(1+3x^{2}+\frac{Q^{2}}{16\mathcal{C}^{2}x^{2}}\right),$		(60)
	$\displaystyle C_{\tilde{Q},\mathcal{V},\mathcal{C}}$	$\displaystyle=$	$\displaystyle T\left(\frac{\partial S}{\partial T}\right)_{\tilde{Q},\mathcal{V},\mathcal{C}}=\frac{8\pi\mathcal{C}x^{2}\left(16\mathcal{C}^{2}x^{2}(3x^{2}+1)-\tilde{Q}^{2}\right)}{16\mathcal{C}^{2}x^{2}(3x^{2}-1)+3\tilde{Q}^{2}}.$		(61)

From Eq. (60), the horizon radius of extremal BH $(T=0)$ can be written as a function of $\tilde{Q}$ and $\mathcal{C}$ as

$\displaystyle x_{\text{ext}}^{2}=\frac{-2\mathcal{C}^{2}+\sqrt{4\mathcal{C}^{4}+3\mathcal{C}^{2}\tilde{Q}^{2}}}{12\mathcal{C}^{2}}.$

(62)

Due to the $T-x$ criticality of charged AdS-BH, thermodynamics of dual CFT also critically change in a similar way as the ratio $\tilde{Q}/\mathcal{C}$ crosses the critical point, which can be determined from the point of inflection as follows

$\displaystyle\left(\frac{\partial T}{\partial x}\right)=0,\ \ \ \text{and}\ \ \ \left(\frac{\partial^{2}T}{\partial x^{2}}\right)=0.$

(63)

Solving these two conditions, we obtain

$\displaystyle x_{c}=\frac{1}{\sqrt{6}},\ \ \ T_{c}=\frac{\sqrt{2}}{\sqrt{3}\pi},\ \ \ S_{c}=\frac{2\pi\mathcal{C}}{3},\ \ \ \frac{\tilde{Q}_{c}}{\mathcal{C}_{c}}=\frac{2}{3}.$

(64)

We examine thermal phase structures of dual CFT in two cases: (I) fixed $\mathcal{C}$ with different values of $\tilde{Q}$ and (II) fixed $\tilde{Q}$ with different values of $\mathcal{C}$. We illustrate the behaviors of $T$ and $C_{\tilde{Q},\mathcal{V},\mathcal{C}}$ as functions of $x$ for the former and latter cases in the first and second rows of Fig. 3, respectively. Note that $x_{\text{max}}$ and $x_{\text{min}}$ are given by

$\displaystyle x_{\text{max,min}}^{2}=\frac{2\mathcal{C}^{2}\mp\sqrt{4\mathcal{C}^{4}-9\mathcal{C}^{2}\tilde{Q}^{2}}}{12\mathcal{C}^{2}}.$

(65)

Two radii $x_{\text{max}}$ and $x_{\text{min}}$ indicate positions of local maximum and minimum of Hawking temperature as shown in Fig. 3 (a) and (c), namely $T_{\text{max}}=T(x_{\text{max}})$ and $T_{\text{min}}=T(x_{\text{min}})$. At the critical value of $\tilde{Q}/\mathcal{C}=2/3$, these two radii coincide at $x_{c}$, which corresponds to the critical temperature $T_{c}$. Remarkably, there exist three thermal states of dual CFT for $\tilde{Q}<\tilde{Q}_{c}$ in case (I) and $\mathcal{C}>\mathcal{C}_{c}$ in case (II). These three states consist of pCFT1 (red), nCFT (green), and pCFT2 (blue), which refers to the states of positive heat capacity when $x_{\text{ext}}<x<x_{\text{max}}$, negative heat capacity when $x_{\text{max}}<x<x_{\text{min}}$, and positive heat capacity when $x>x_{\text{min}}$, respectively. It is important to note that our notation might cause some confusion when $x_{\text{max}}$ is less than $x_{\text{min}}$. As defined above, $x_{\text{max}}$ and $x_{\text{min}}$ in this paper refer to the points at which the temperature $T$ is at its maximum and minimum, respectively. Please do not be confused by these terms.

Using the Maxwell’s equal area law, the curve associated with the unstable nCFT phase in the $T-x$ plane in cases (I) and (II) can be replaced by a horizontal line, which indicates the first-order phase transition between these pCFT1 and pCFT2. In Appendix B, we elaborate on the detailed calculations using the method employed by [77] to obtain the Hawking-Page phase transition temperature $T_{f}$ within the holographic thermodynamics framework. The temperatures for cases I and II are as follows:

$\displaystyle t^{*}=\frac{3+\sqrt{3(q-3)(q-1)}-q}{2\sqrt{3+\sqrt{3(q-3)(q-1)}-2q}},$

(66)

and

$\displaystyle t^{*}$

$\displaystyle=$

$\displaystyle\frac{3c+\sqrt{3(1-3c)(1-c)}-1}{2\sqrt{3c^{2}+c\sqrt{3(1-3c)(1-c)}-2c}},$

(67)

respectively, where $t^{*}=T_{f}/T_{c},q=\tilde{Q}/\tilde{Q}_{c}$ and $c=\mathcal{C}/\mathcal{C}_{c}$. The horizon radii for Small and Large BHs of case I and II are expressed respectively as follows

	$\displaystyle\frac{x_{S}}{x_{c}}=\frac{q}{\sqrt{3+\sqrt{3(q-3)(q-1)}-2q}},$		(68)
	$\displaystyle\frac{x_{L}}{x_{c}}=\sqrt{3+\sqrt{3(q-3)(q-1)}-2q},$		(69)

and

	$\displaystyle\frac{x_{S}}{x_{c}}=\frac{1}{\sqrt{3c^{2}+c\sqrt{3(1-3c)(1-c)}-2c}},$		(70)
	$\displaystyle\frac{x_{L}}{x_{c}}=\frac{1}{c}\sqrt{3c^{2}+c\sqrt{3(1-3c)(1-c)}-2c}.$		(71)

Note that we summarize our numerical results of some important parameters in our study in Table 1.

The thermodynamic potential associated with fixed $(\tilde{Q},\mathcal{V},\mathcal{C})$ ensemble is the Helmholtz free energy

$\displaystyle F\equiv E-TS=\frac{\mathcal{C}x^{n-2}}{R}\left(1-x^{2}+\frac{(2n-3)}{4\eta^{2}(n-1)^{2}\mathcal{C}^{2}}\frac{\tilde{Q}^{2}}{x^{2n-4}}\right).$

(72)

The variation of $F$ reads

	$\displaystyle dF$	$\displaystyle=$	$\displaystyle dE-TdS-SdT,$		(73)
		$\displaystyle=$	$\displaystyle(TdS+\tilde{\Phi}d\tilde{Q}-pd\mathcal{V}+\mu_{\mathcal{C}}d\mathcal{C})-TdS-SdT,$
		$\displaystyle=$	$\displaystyle-SdT+\tilde{\Phi}d\tilde{Q}-pd\mathcal{V}+\mu_{\mathcal{C}}d\mathcal{C},$

where we have used Eq. (50) for $dE$, this yields

$\displaystyle\left(\frac{\partial F}{\partial T}\right)_{\tilde{Q},\mathcal{V},\mathcal{C}}=-S,\ \ \ \left(\frac{\partial F}{\partial\tilde{Q}}\right)_{T,\mathcal{V},\mathcal{C}}=\tilde{\Phi},\ \ \ \left(\frac{\partial F}{\partial\mathcal{V}}\right)_{T,\tilde{Q},\mathcal{C}}=-p,\ \ \ \left(\frac{\partial F}{\partial\mathcal{C}}\right)_{T,\tilde{Q},\mathcal{V}}=\mu_{\mathcal{C}}.$

(74)

As pointed out in [32], there is no critical point found in the $p-\mathcal{V}$ plane in the dual CFT. This result from the holographic thermodynamics is different from the $P-V$ criticality behavior from the extended phase space approach as discussed in last subsection. Therefore, we will consider $F$ as a function of $T$ instead of $p$ with constant $\tilde{Q}$ or $\mathcal{C}$ to investigate the global stability of dual CFT. By considering $x$ as a parameter, we parametrically plot between $F$ in Eq.(72) and $T$ in Eq.(55) for dual CFT in the case (I) and (II) as shown in the left and right panels of Fig 4, respectively. There are cusp points in $F(T)$ curve where $C_{\tilde{Q},\mathcal{V},\mathcal{C}}$ diverge. At these points, there are the second-order phase transitions occuring, manifested from the fact that $\left(\frac{\partial^{2}F}{\partial T^{2}}\right)_{\tilde{Q},\mathcal{V},\mathcal{C}}=-\frac{C_{\tilde{Q},\mathcal{V},\mathcal{C}}}{T}$, which can be derived via the first relation in Eq. (74).

We will discuss about the phase transition of dual CFT via its $F(T)$ curve in more details as follows. First, let us consider in the case (I). We depicted the $F(T)$ curves when $\tilde{Q}<\tilde{Q}_{c}$, $\tilde{Q}=\tilde{Q}_{c}$ and $\tilde{Q}>\tilde{Q}_{c}$ in figures (a), (b) and (c) of Fig 5. The free energy shows the swallowtail shape for $\tilde{Q}<\tilde{Q}_{c}$. In this case, the pCFT1 phase is thermodynamically preferred in low temperatures, i.e. both in the region of $T<T_{\text{min}}$ and $T_{\text{min}}\leq T<T_{f}$. There is a first-order phase transition from pCFT1 to pCFT2 occurring at $T_{f}$. At slightly higher than $T_{f}$, the pCFT2 phase becomes the most thermodynamically preferred since it has the lowest free energy. The swallowtail shape of $F(T)$ curve with $\tilde{Q}=\tilde{Q}_{c}$ shrink to appear as a kink at $T_{c}$, where the transition between pCFT1 and pCFT2 phases become the second-order type of phase transition. For $\tilde{Q}>\tilde{Q}_{c}$, there is only one phase with continuous curves with positive heat capacity for any $T$, thus no phase transition occurs.

We plot $F$ versus $T$ in the case (II) in Fig 5 (d), (e) and (f) from small to large values of $\mathcal{C}$. The results indicate that phase structures of dual CFT in case (II) are qualitatively similar to the case (I). Namely, $F(T)$ displays the swallowtail shape of three phases for $\mathcal{C}>\mathcal{C}_{c}$, the pCFT1-pCFT2 first-order phase transition occurs at $T=T_{f}$. At the critical value of $\mathcal{C}$, the nCFT disappears and the transition between pCFT1 and pCFT2 becomes the second-order phase transition at $T_{c}$. For the small values of $\mathcal{C}$, there is only one thermal state of CFT with positive heat capacity at any value of $T$.