Critical phenomena of static charged AdS black holes in conformal gravity

The extended thermodynamics of static charged AdS black holes in conformal gravity is analyzed. The $P-V$ criticality of these black holes has some unusual features. There exists a single critical point with critical temperature $T_c$ and critical pressure $P_c$. At fixed $T>T_c$ (or at fixed $P>P_c$), there are two zeroth order phase transition points but no first order phase transition points. The systems favors large pressure states at constant $T$, or high temperature states at constant $P$.

The reentrant phase transition includes both first and zeroth order phase transitions. However, the origin of zeroth order phase transition is still unclear. This makes it interesting to consider black hole systems containing only the zeroth order phase transition. The thermodynamics of the AdS black holes in conformal gravity [28] to be studied in this work provides precisely such an example.
On the other hand, although taking the cosmological constant as a thermodynamic pressure is nowadays a common practice, such operations implicitly assume that gravitational theories which differ only in the values of the cosmological constants are considered to fall in the "same class", with unified thermodynamic relations. The common excuse for doing this is that the classical theory of gravity may be an effective theory which follows from a yet unknown fundamental theory, in which all the presently "physical constants" are actually moduli parameters that can run from place to place in the moduli space of the fundamental theory. Given that the fundamental theory is yet unknown, it is preferable to consider the extended thermodynamics of gravitational theories involving only a single action, which requires that all variables included in the thermodynamical relations must be integration constants. It so happens that, in conformal gravity, the cosmological constant comes as an integration constant rather than as a bare parameter which appear explicitly in the classical action.
In this paper, we consider the P − V criticality of the static charged AdS Black hole in conformal gravity. There exists a single critical temperature, above which there are two zeroth order phase transition points. In the next section, we first revisit the thermodynamics of static charged AdS black hole in conformal gravity in the extended phase space. Section 3 is devoted to the P − V criticality, particular attention is paid toward the appearance of critical points and the zeroth order phase transitions. Finally, some concluding remarks are given in the last section.
2 Thermodynamics of static charged AdS Black hole in conformal gravity revisited We consider the static charged AdS Black hole in conformal gravity. The action of conformal gravity reads where the unusual sign in front of the Maxwell term is inspired by critical gravity [29] and is required by requiring that the Einstein gravity emerges from conformal gravity in the infrared limit [30].
The static black hole solution with AdS asymptotics for conformal gravity is found in [28], which takes the form where and is the corresponding Maxwell field. dΩ 2 2, represents the line element of a 2d maximally symmetric Einstein space with constant curvature 2 , with = 1, 0 and −1, respectively. There are six parameters Q, c 0 , c 1 , d, Λ, in the solution, five of which are integration constants, and the last one determines the spacial sectional geometry of the horizon. These parameters obey a constraint so there are actually only 5 independent parameters (c 0 is to be considered to be determined by other parameters as in (5)). Except the discrete parameter , the rest 4 parameters Q, c 1 , d, Λ are related to conserved charges: electric charge, charge of massive spin-2 hair, enthalpy and pressure, respectively. However, at fixed charges Q, c 1 , d, Λ and , there still exist a discrete freedom in choosing the integration constant c 0 : Under the point of view of taking the cosmological constant Λ as a thermodynamic the energy calculated by employing the Noether charge associated with the time-like Killing vector [31] should be identified with the enthalpy H of the gravitational system.

It reads
where r 0 > 0 denotes the largest real root of f (r) which corresponds to the event horizon black hole 1 , α is overall coupling which is present in the action (1). Because of the double-valuedness of the integration constant c 0 , one immediately sees a doublevalued behavior of the enthalpy. Such behaviors are also present in the four dimensional charged rotating black hole [32] and six dimensional static black holes [33] of conformal gravity. One can expect that the temperature and Gibbs free energy may also be double-valued. These double-valued variables must be all considered in order to have a holistic look at the thermodynamics in the extended phase space of the black hole.
The thermodynamical conjugate of the pressure, i.e. the "thermodynamic volume", is given by It can be seen that the sign of V is determined by the sign of the parameter d. Besides P and V , all the other thermodynamic quantities are given in [28]. The temperature is 1 We take Λ < 0, and so there is no cosmological horizon in the solution. and its conjugate, i.e. the entropy is The electric charge and the conjugate potential are respectively The parameter c 1 is a massive spin-2 hair which is now taken as a novel dimension in the thermodynamic phase space. We label this novel dimension and its conjugate as Ξ, Ψ: Throughout this work, we will take the normalization α = 2. It can be checked that the first law of thermodynamics dH = T dS + Φ dQ e + Ψ dΞ + V dP (15) and the Smarr relation hold in the extended thermodynamic phase space [28]. The absence of T S and ΦQ terms in the Smarr relation can be explained by scaling arguments. The enthalpy can be viewed as a homogeneous function of the extensive variables S, Q e , Ξ, P , i.e.
Assuming each extensive variable has a scaling dimension which is denoted respectively. If the enthalpy H itself has scaling dimension d H , then after a rescaling of the extensive variables we get Taking the first derivative with respect to λ and then setting λ = 1, we get  (16), as expected.
While considering critical behaviors, the Gibbs free energy will play an important role. It is given as follows: On the other hand, the Helmholtz free-energy can be obtained from the Euclideanized action: A direct check yields This in turn justifies the explanation of H as thermodynamic enthalpy.
Before proceeding, let us reveal a natural constraint of the black hole solutions.
From f (r 0 ) = 0 we get P = − 24π(r 0 2 c 1 +c 0 r 0 +d) . On the other hand, from eq. (5) we get . Inserting both into (9), one can obtain which leads to the constraint on T and r 0 This constraint implies an upper bound of T at fixed horizon radius r 0 , or an upper bound of r 0 at fixed temperature T .

Critical points
To consider the P −V criticality of the black hole, we should begin with the equation of state (EOS) in P − V plane at fixed conserved charges Q and c 1 . The EOS arises from the expression (9) for the temperature T . However, to use (9) as a reasonable EOS, we need to eliminate the parameters c 0 and d. Assuming that both of these parameters are nonzero. Then the condition f (r 0 ) = 0 yields Inserting (21) into eqs. (5) and (9) respectively, one gets Then from eq.(23), we find d = 16 3 π P r 3 0 − 4 T π r 2 0 + c 1 r 2 0 .
Solving this equation for P we get and thanks to the condition (20), both branches of solutions should be considered physical. Comparing the above expressions for the pressure with the van der Waals we are tempted to use the variable v = 2r 0 as an effective specific volume for the black hole system under consideration. Thus we rewrite the full EOS as The critical points (P c , v c , T c ) results from the conditions The partial derivative in (26) and (27) can be evaluated directly using (25), which read Solving these two equations and substituting into (28), we get two sets of critical point parameters as follows.
In the above, X can take two discrete values X 1 or X 2 : However, the physical critical point must obey the constraints P c > 0, r c > 0 and T c > 0, which lead to This exclude the choice c 1 < 0 and X = X 1 , and we are left with a single critical point which is characterized by the parameters in which c 1 > 0. These parameters satisfy the relation which gives a pure number and is independent of all parameters. Actually, this relation is very similar to the one of the van der Waals system, which behaviors as Pcvc Tc = 3 8 at its critical point. However, if one replaces v in Eq.(35) by the thermodynamic volume V , the result of this relation will be no longer independent of parameters. In this sense, it is more natural to consider v as the specific volume instead of the thermodynamic volume V . Note that the existence of critical point also exclude the possibility of c 1 = 0. Also, the square sum of the charge Q and the signature must also be nonzero.
Therefore, there is neither need to distinguish the case Q = 0 from the Q = 0 cases (as long as = 0) nor need to distinguish the = 0 case from the = ±1 cases (as long as One may also be curious about the cases with d = 0 (the BPS black hole) or c 0 = 0, both of which will result in an EOS of ideal gas after considering eq. (9) directly. Thus they are all out of our discussion. Another degenerated case c 1 = 0, Q = 0 corresponds to the Schwarzschild-AdS black hole. In this case one can never find physical critical points as is known in [7].
The isothermal plots at generic parameters Q, c 1 are depicted in Fig.1 (the right plot is a magnification of a single isotherm at the temperature T = 1.5T c ). While creating the plots, we take , c 1 and c 2 1 √ Q 2 + 2 respectively as units for v c , T c and P c . The pressure corresponding to the extremal specific volume is denoted P 1 .
It can be seen that on each isotherm there is an upper bound for the specific volume (black hole radius) v ex . For v < v ex , the isotherm can be subdivided into two segments, i.e. the lower branch and the upper branch, which reflect the double-valuedness of the pressure. The difference between the T < T c and T > T c curves lies in that, each branch of the isotherm in the former case is monotonic with respect to the special volume v, while the lower branch in the latter case is non-monotonic. Consequently phase transitions will occur only in the T > T c regime.
Another way of subdividing the isotherms is according to the sign of ∂P ∂v T , which is inversely proportional to the isothermal compressibility According to the sign of the isothermal compressibility, each isotherm with T > T c can be subdivided into 4 segments, two with positive isothermal compressibility and two with negative isothermal compressibility.
Let us take a closer look at the magnified plot given on the right diagram in Fig.1.
This is a curve corresponding to isotherm with T = 1.5T c . The lower and upper branches of the curve is joined together at the point D which corresponds to the extremal specific volume v ex and the pressure P = P 1 . On the lower branch (plotted in solid line) one can see that there is a local maximum P 2 and local minimum P 0 for P , the corresponding points on the isotherm are marked with B and C respectively.
The 4 segments of the isotherm are IB , BC , CD and DJ respectively. Among these, IB and CD have negative isothermal compressibilities which imply that black hole states falling in these segments may be unstable. The right plots in Fig.1 is typical for all T > T c . At such temperatures, we can subdivide the range of the pressure into 4 regimes: P < P 0 , P 0 ≤ P < P 1 , P 1 ≤ P < P 2 and P ≥ P 2 . When P < P 0 , there is a single unstable black hole phase represented by the segment IM . Due to its unstable nature, a perturbative increase in the pressure would result in an increase of the black hole radius until it reaches the state M, then through a phase transition it will enter the second regime for the pressure, P 0 ≤ P < P 1 .
In this second regime for the pressure, there are 3 different black hole states for a single pressure value, among these, only the one with intermediate sized specific volume (i.e. the state lying on the segment KC ) is stable. Therefore, as the pressure increases, the black hole state will evolve from the state C until the state K is reached and then the pressure enters the next regime, P 1 ≤ P < P 2 . There are still 3 black hole states at each pressure values in this regime, however, two of these have positive isothermal compressibility (i.e. the states lying on the KB and DN segments), so it is hard to tell which is more stable by looking at the EOS alone. If the pressure enters the fourth regime, P ≥ P 2 , then there is only a single stable phase which corresponds to states on the segment NJ . From the above analysis, it is clear that there are two possible phase transition points, the first one occurs at P = P 0 , where the black hole will most probably transit from the state M to the state C which is more stable upon perturbation. The second phase transition point occurs either at P = P 1 (if the Gibbs free energy on the segment KB is higher than that on DN ) or at P = P 2 (if the Gibbs free energy on the segment KB is lower than that on DN ). In the next subsection, it will be clear that the Gibbs free energy on the segment KB is always lower than it is on DN , so the second phase transition point occurs at P = P 2 , where the black hole picks the state B instead of N, because the state B has lower Gibbs free energy than the state N .

Gibbs free energy and the zeroth order phase transitions
In order to have a further look at the critical points, we need to plot the Gibbs free energy versus pressure at fixed temperature. The Gibbs free energy G as a function of T and P is quite complicated and cannot be given explicitly with ease. So we will try to present the G − P relationship in terms of a pair of parametric equations.
First we can invert the relation (23) to get P as a function of T and other parameters.
Eliminating c 0 in the resulting expression by use of (21), we get where we have also replaced r 0 by v/2. Inserting this equation together with (21) into (17), we get In (1) and (37), d cannot be taken as a free parameter, because there is an extra constraint (22), which can be rewritten as It can be seen in Fig.2 that, for all T > T c , a downcast swallow tail appears on each G − P curve. From T = T c and downwards, the swallow tail disappears, with T = T c corresponding to the critical point. This is different from that of a Van der Waals liquid-gas system, where T < T c is required for a phase equilibrium. The same phenomenon has also been observed in the study of criticality associated with the GB coupling constant [25] in Gauss-Bonnet gravity.
The downcast swallow tail is different from the usual upcast one, which corresponds to first order phase transitions [7]. On the magnified plot given on the right plot of Fig.2, it is clear that the Gibbs free energy on the segment CKB is lower than that on CDN and BLM . At P = P 0 and P = P 2 , there exist discontinuities for the Gibbs free energy, indicating that there are zeroth order phase transitions at these two particular pressures.
If one follows the red curve given in the left plot of Fig.2, it would be clear that at sufficiently high pressure, the Gibbs free energy on the dotted segment would eventually become lower than it is on the solid segment. This implies that the thermodynamics of the system favors large pressure states. On the other hand, if one looks at the Gibbs free energy versus temperature plots at fixed pressure (Fig.3), one would see the similar downcast swallow tail at P > P c . Moreover, following a single G − T curve reveals that the system favors high temperature states if the pressure is kept fixed.

Concluding remarks
In this paper, we considered the P − V criticality of static charged AdS black holes in conformal gravity. Unlike the cases of Einstein gravity, the cosmological constant arises as an integration constant in conformal gravity, making the analysis for P − V criticality more self-contained, e.g. without need to consider systems with different actions.
The thermodynamics in the extended phase space for black hole in conformal gravity possesses several unusual features: • there exists only one critical point but there are two phase transition points, both of which corresponds to zeroth order phase transitions; • there is no first order phase transition in the system; • the phase transition can occur only when T > T c (or P > P c ) but not the other way round; • at fixed T > T c , the system favors large pressure states, whilst at fixed P > P c , the system favors high temperature states.
We do not know of any other black hole or ordinary matter systems which exhibit similar thermodynamic behaviors. It would be interesting to find other examples which yield similar behaviors, because otherwise the system under study would seem to be too bizarre to understand.