${\cal P}-v$ Criticality in Gauged Supergravities

AdS black holes show richer transition behaviors in extended phase space by assuming the cosmological constant and its conjugate quantity to behave like thermodynamic pressure and thermodynamic volume. We study the extended thermodynamics of charged dilatonic AdS black holes in a class of Einstein-Maxwell-dilaton theories that can be embedded in gauged supergravities in various dimensions. We find that the transition behaviors of higher dimensional dilatonic AdS black holes are different from the four dimensional counterparts, and new transition behaviors emerges in higher dimensions. First, there exists standard Van der Waals transition only in a five dimensional dilatonic AdS black hole with two equal charges. Second, there emerge a new phase transition branch in negative pressure region in six and seven dimensional dilatonic black holes with two equal charges. Third, there emerge transition behaviors in higher dimensional black hole with single charge cases, which are absent in four dimensions.


Introduction
In view of anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1][2][3], the strongly coupled boundary conformal field theory (CFT) can be understood via anti-de Sitter (AdS) black holes.In the dictionary of AdS/CFT correspondence, Hawking-Page phase transition [4] of Schwarzchild AdS black hole was well interpreted by Witten [5] as a confinement/deconfinment phase transition in the boundary CFT.However, the dual CFT of the phase transition, Van der Waals transition, of Reissner-Nordström (RN) AdS black hole is still unclear.So it is valuable to study the phase transitions deeply in both boundary and bulk views.The phase transitions of charged AdS black hole systems have been studied in literatures, e.g.[6][7][8][9], for many years.Recently, the idea of treating cosmological constant and its conjugate quantity as thermodynamic pressure and thermodynamic volume in first law of black hole thermodynamics were studied in Refs.[10,11] and then the phase transition in extended phase space was considered in Ref. [12].In the frame of extended phase space, many interesting new features appear, for examples, λ-line transition [13], reentrant phase transitions [14,15], triple points [16], special isolated critical point [17] and so on [18,19].
We refer to e.g.[20] and references therein for more details of this subject.
In the transitions mentioned above, the effect of scalar fields are not considered.And one of the simplest scalar modification of Einstein-Maxwell (EM) theory is so called Einstein-Maxwell-Dilaton (EMD) theory.There are many different EMD theories due to different dilaton coupling constants and scalar potentials, and phase transitions are studied in these theories, e.g.[21][22][23].In our work, we focus on a class of EMD theory inspired by supergravity.
The extremal RN black holes in supergravity can be viewed as one of the bound states of the basic U (1) building blocks with zero binding energy [24,25].On the other hand, while EM theories can be embedded in string and M-theory in four and five dimensions only, charged dilatonic AdS black holes in gauged supergravities [26][27][28][29] can be embedded in higher dimensions.
Actually, according to Ref. [30], there exist different phase transitions for charged dilatonic AdS black holes with different dilaton coupling constants in EMD theory with stringinspired potential in four dimension.So it is natural to exploring transition behaviors in higher dimensional supergravities for different dilaton coupling constants.
The rest of this paper is organized as follow.In section 2, we review the thermodynamics of charged dilatonic AdS black holes in gauged supergravity.In section 3, we study the phase transitions of charged dilatonic AdS black holes in extended phase space of canonical ensemble in diverse dimensions and dilaton coupling constants.We conclude in section 4.

Thermodynamics of charged dilatonic AdS black holes
The Lagrangian of general EMD theory consisting gravity, single Maxwell field A and a dilaton field φ in D ≥ 4 dimensions is given by where e = √ −g, F = dA, V is scalar potential inspired by gauged supergravity and a is the dilaton coupling constant which can be reparameterized by [31] where ã is another dilaton coupling constant appearing in the potential V and N is also used to represent dilaton coupling constant.The charged dilatonic AdS black holes with a can be viewed as dilatonic AdS black hole with N equal charges [24,25].So the value of N should be positive integers required by supergravity.The reality condition of (a, ã) requires that When N = N RN , i.e., a = 0, the dilaton decouples and the theory reduces to EM theory.
Although the Lagrangian can be made real by letting φ → iφ when N > N RN , we shall not consider such situation at all.Due to the maximal dimension allowed by supergravity is seven, so we can easily present all the possible values of N in different dimensions in where g is gauge coupling constant (there should be no confusion between the gauge coupling constant and the determinant of the metric).
The static AdS black hole solutions for Lagrangian (2.1) with scalar potential (2.4) and constraints (2.2) are given by [31] ds ) where parameters m and q characterise mass and electric charge, and dΩ 2 D−2 represents the unit (D − 2)-sphere, (D − 2)-torus or hyperbolic (D − 2)-space.The topological black holes can be easily obtained by some appropriate scaling.In our work, we only consider spherical black holes for simplification.
The event horizon of black hole is determined by the largest (real) root of f (r 0 ) = 0.
And the thermodynamic quantities are given by [31] where Γ indicates the Gamma function, and (M, T, S, Q, Φ) denote mass, temperature, Bekenstein-Hawking entropy, electric charge and electric potential respectively.Further, the gauge coupling constant (cosmological constant) can be interpreted as thermodynamical pressure P in the extended phase space, The corresponding thermodynamic volume is given by (2.12) The thermodynamic volume satisfies the "Reverse Isoperimetric Inequality" conjecture [11].
Although the conjecture is not proven, it has been checked in most black holes and follows from null-energy condition [32].These quantities satisfy the following first law and Smarr relation For our purpose to study the P − V phase transitions of the charged dilatonic AdS black holes in the canonical ensemble of extended phase space, we fix the electric charge Q and treat it as not a thermodynamical variable for remainder discussion.So the mass M can be viewed as enthalpy H, In order to study the P − V phase transitions, it is convenient to introduce a special volume v by analysing the dimensional scaling [12].By assuming the shape of black hole is regular sphere, the special volume [14] can be viewed as the effective radius of black hole, (2.16) So we can study the P − v criticality instead of P − V criticality in reminder discussions.

P − v criticality
The P − v phase transitions behaviors of four dimensional charged dilatonic AdS black holes with N = 1, 2, 3, 4, which can be viewed as STU black holes with single charge, two equal charges, three equal charges and four equal charges (or RN AdS black hole), have been studied in [30].The behaviors of five dimensional charged dilatonic AdS black hole with N = 3, which can be viewed as RN AdS black hole, have also been studied in [14].
So we focus on N = 1 and N = 2 cases in five, six and seven dimensions.It is convenient to study the behaviors according to the different values of N , because the cases with same value of N have similar behaviors.However, the transition behaviors also can be changed by dimensional parameter.
3.1 N = 2 cases D = 5 case: As shown in Ref. [30], the thermodynamic pressure goes to zero as the thermodynamic volume decreases to zero at sufficiently low temperatures for the four dimensional charged dilatonic AdS black holes with N = 2.As the temperature increases, there is a phase transition at some critical temperature and the thermodynamic pressure goes to infinity as the thermodynamic volume decreases to zero in the system at sufficiently high temperature.The behavior in five dimension is rather different from the four dimensional counterpart.From the analysis of previous section, it is hard to obtain the analytic expression of equation of state because of the coupling of different thermodynamic quantities.
However, we can analyse the phase transitions by numeric method.We plot the P − v and T − v diagrams in Fig. 1.The corresponding critical point can be obtained by The critical point and the universal ratio independent of Q are given by From Fig. 1, we find the P − v transitions of five dimensional charged dilatonic AdS black holes with N = 2 have Van der Waals behaviors which are similar to five dimensional RN AdS black hole [14].Actually, the four dimensional charged dilatonic AdS black holes with N = 3 also have similar behaviors [30].However, we study the P − v transitions for all possible values of N supported by gauged supergravities, and the similar behaviors do not appear again.For more details about the standard Van der Waals transitions of these cases, we refer to our future work [33].2. It is exciting that one of the critical Table 2: Two critical points with N = 2 in six and seven dimensions.
pressure is negative and the behavior was not obtained in previous related researches to our knowledge.It can be explained analogously as the P − v phase transitions of black hole systems undergo the repulsive force on the horizon.Actually, this similar interesting phase transition phenomena at negative pressure have been observed in liquid-liquid system, for examples, [34].On the other hand, at the negative critical pressure, all of other quantities of black holes are positive, such as mass, entropy, temperature and so on.So the negative pressure is reasonable.The scaling of the critical point at negative pressure is also same with its positive counterpart.The ratios independent of Q in both cases which are given by D = 6 : Because of the Van der Waals-like behaviors are similar in six and seven dimensions (the similar behavior is also appeared in D > 7 dimensions which are not supported by gauged supergravity theory, so we do not consider them in this paper), so we only plot P − v diagrams with N = 2 of six dimensional charged dilatonic AdS black hole which is given by Fig. 2.There are two branches of isotherms in the P − v diagrams at a certain temperature.
At sufficiently high temperature, the upper branch of isotherms undergoes an normal Van der Waals phase transition which also appears in RN AdS black holes.As the temperature decreases, two branches intersect and exchange part of their lines.Then at sufficient low temperature, the lower branch of isotherms also undergoes a Van der Waals-like behavior.
It should mention that the lower branch of isotherm will be disappeared at sufficiently low temperature.The thermodynamic pressure goes to positive infinity as the thermodynamic volume goes to zero.The thermodynamic pressure goes to zero as the thermodynamic volume goes to infinity.
We find there exists another interesting feature, the incompressible points, in the lower We also plot the isobars of T − v diagrams in Fig. 3 which provides another perspective to study the Van der Waals-like phase transition.The Gibbs free energy can be obtained simply by G ≡ G(T, P) = M − T S. According to [12], we also plot the isobars of G − T diagrams in Fig. 4. From Fig. 4, we find there also exist swallowtail behavior at negative pressure which is a typical feature characterize the multivalued function (also can be appeared in many other systems, e.g.[36]).The swallowtail disappears at both positive and negative critical pressures.The critical exponentials for the critical point with positive critical pressure are given by which is same with Van der Waals behaviors.So we can definitively claim the behavior under the extended phase space assumption with positive critical pressure is Van der Waals  behaviors.We also calculate the critical exponentials in the negative critical pressure.The values of α, β and δ are the same.However the isothermal compressibility κ T = − 1 v ∂v ∂P | T < 0 which can not be used to obtain γ.If taking the absolute value of κ T , the value of γ is given by 3. We plot the P − T plane in Fig. 5.The Van der Waals-like behavior in the negative pressure can be also viewed as small/large black hole transitions.
Here we give some discussions about the new transition behavior mentioned above.
First of all, we determine the black hole with the new transition exists indeed.During the numerical analysis, horizon radius r 0 should be the outer horizon rather than inner horizon.
In the negative pressure region, AdS black holes convert to de Sitter (dS) black holes.For dS black holes, there exist another cosmological horizon larger than event horizon.From Fig. 6 , the first intersection of −g tt and g −1 rr is the outer horizon and the second one is the cosmological horizon.So we can say the black holes with the strange transition behaviors exist indeed.Actually, the transition behavior of the critical point is similar to the phase transition in RN-dS black hole which is briefly given in Appendix A. So the phase transition may not be connected with AdS/CFT correspondence.And the boundary field interpretation should be studied further.Second, all the extended thermodynamical analysis is based on assuming the cosmological constant as thermodynamic pressure.However, there is no concrete evidence to prove the thermodynamic pressure is the real pressure.
Therefore, the conjugate thermodynamic volume maybe not real definition of black holes volume.The black hole volume should be studied further.Third, under the extended phase space assumption, the thermodynamic volume of dilatonic AdS black holes are not regular sphere topology in the sense of horizon radius and usually leads to unusual P − v transitions, e.g.[15,16].Last, the negative critical pressure appears in higher dimensions only.Through dimensional reduction [37,38], the new phase transition of low dimensional charged dilatonic black hole disappears.

N = 1 cases
Now we study the P − v phase transitions of charged dilatonic AdS black hole with N = 1 in five, six and seven dimensions.The N = 1 supergravities can be viewed as Kaluza-Klein theory.As shown in Ref. [30], there is no Van der Waals transition in four dimension.The thermodynamic pressure goes to negative infinity as the volume decreases to zero which is same with Hawking-Page transition.However, the behavior is different in higher dimensions.
We can obtain the critical points from Eq. (3.1) and the universal ratios are given by Table 3.
We plot the P − v diagram of N = 1 in five and six dimensions in Fig. 7 (the behavior in D > 6 is similar to the six dimensional counterpart, so we do not plot them here).We find the thermodynamic pressure does not goes to positive infinity as the thermodynamic volume decreases to zero.However, there exist Van der Waals-like transition behaviors in    another one is new behavior in which the critical pressure is negative.In the new branch transition, there exist an incompressible point.Through the isothermal compressibility of the critical point is not compatible with ordinary thermodynamics, it is still significant for the studying of the new phase transitions in both boundary and bulk view of gauge / gravity duality.Third, we find there exist Van der Waals -like transitions in N = 1 case in D > 4 dimesions which is absent in four dimension.
Due to the rich phase behaviors of charged dilatonic AdS black holes, it is valuable to study the extended thermodynamics of more general AdS black holes, such as rotating [39] or dyonic [40,41] black holes in (ω-deformed) gauged supergravity systems [42,43] further.

TFigure 1 :
Figure 1: P−v diagrams (left) and T −v diagrams (right) of charged dilatonic AdS black hole with N = 2, D = 5.We set electric charge Q = 10.In P − v diagrams, the lines represent isotherms with temperatures 3T c /2, T c , T c /2 from top to bottom.In T − v diagrams, the lines represent isobars with pressures 3P c /2, P c , P c /2 from top to bottom.
= 6 and D = 7 cases: Now we study the P − v transitions in six and seven dimensions.Unlike the counterparts in D = 4 and D = 5, The charged dilatonic AdS black holes in six and seven dimensions have some new Van der Waals-like behaviors.By solving Eq. (3.1), we obtain two critical points, which are given in Table

Figure 2 :
Figure 2: Isotherms in P −v diagram of charged dilatonic AdS black hole with N = 2, D = 6.We set electric charge Q = 10.The two black points represent the critical points and the two blue points represent incompressible points.

Figure 3 :
Figure 3: T − v diagrams of dilatonic black hole with N = 2, D = 6.We set electric charge Q = 10.The two black points represent the critical points.

Figure 4 :
Figure 4: G − T diagrams of charged dilatonic AdS black hole with N = 2, D = 6.We set electric charge Q = 10.The black points represent the critical points.

Figure 5 :
Figure 5: P − T plane of dilatonic black hole with N = 2, D = 6.We set electric charge Q = 10.In the plane, the top and bottom lines represent the positive and negative pressure zones.The black points represent the critical points.

Figure 6 :
Figure6: The horizontal axis is r 0 , Blue line and orange line are −g tt and g −1 rr of charged dilatonic black hole for N = 2 in six dimension at negative critical point respectively.

Figure 7 :
Figure 7: P − v diagrams of charged dilatonic AdS black hole for N = 1 in five and six dimensions.We set electric charge Q = 10.The temperature of lines from top to bottom are T = 1.2 T c , T c , and 0.7 T c respectively.

Table 1 .
The potential can be expressed in terms of a super potential W ,

Table 1 :
All possible values of N for gauged supergravities in diverse dimensions.

Table 3 :
Critical points with N = 1 in five, six and seven dimensions.