Phase transitions, geometrothermodynamics and critical exponents of black holes with conformal anomaly

We investigate the phase transitions of black holes with conformal anomaly in canonical ensemble from different perspectives. Some interesting and novel phase transition phenomena have been discovered. Firstly, we discuss the behavior of the specific heat and the inverse of the isothermal compressibility. It is shown that there are striking differences in Hawking temperature and phase structure between black holes with conformal anomaly and those without it. In the case with conformal anomaly, there exists local minimum temperature corresponding to the phase transition point. Phase transitions take place not only from an unstable large black hole to a locally stable medium black hole but also from an unstable medium black hole to a locally stable small black hole. Secondly, we probe in details the dependence of phase transitions on the choice of parameters. The results show that black holes with conformal anomaly have much richer phase structure than those without it. There would be two, only one or no phase transition points depending on the parameters we have chosen. The corresponding parameter region are derived both numerically and graphically. Thirdly, geometrothermodynamics are built up to examine the phase structure we have discovered. It is shown that Legendre invariant thermodynamic scalar curvature diverges exactly where the specific heat diverges. Furthermore, critical behaviors are investigated by calculating the relevant critical exponents. It is proved that these critical exponents satisfy the thermodynamic scaling laws, leading to the conclusion that critical exponents and the scaling laws can reserve even when we consider conformal anomaly.


Introduction
Black hole thermodynamics has long been one of exciting and challenging research fields ever since the pioneer research made by Bekenstein and Hawking [1]- [2]. A variety of thermodynamic quantities of black holes has been studied. In 1983, Hawking and Page [3] discovered that pure thermal radiation in AdS space becomes unstable above certain temperature and collapses to form black holes. This is the well-known Hawking-Page phase transition which describes the phase transition between the Schwarzschild AdS black hole and the thermal AdS space. This phenomenon can be interpreted in the AdS/CFT correspondence [4] as the confinement/deconfinement phase transition of gauge field [5]. Since then, phase transitions of black holes have been investigated from different perspectives. For recent papers, see [6]- [26].
One of the elegant approach is the thermodynamic geometry method, which was first introduced by Weinhold [27] and Ruppeiner [28]. Weinhold proposed metric structure in the energy representation as g W i,j = ∂ i ∂ j M (U, N a ) while Ruppeiner defined metric structure as g R i,j = −∂ i ∂ j S(U, N a ). These metric structures are respectively the Hessian matrix of the internal energy U and the entropy S with respect to the extensive thermodynamic variables N a . And Weinhold's metrics were found to be conformally connected to Ruppeiner's metrics through the map dS 2 R = dS 2 W T [29]. Ruppeiner's metric has been applied to investigate various thermodynamics systems for its profound physical meaning. For more details, see the nice review [30]. Recently Quevedo et al. [31] presented a new formalism called geometrothermodynamics, which allows us to derive Legendre invariant metrics in the space of equilibrium states. Geometrothermodynamics presents a unified geometry where the metric structure describes various types of black hole thermodynamics [32]- [41].
Apart from the thermodynamic geometry, critical behavior also plays a crucial role in the study of black hole phase transitions. The critical points of phase transitions are characterized by the discontinuity of thermodynamic quantities. So it is important to investigate the behavior in the neighborhood of the critical point, especially the divergences of various thermodynamic quantities. In classical thermodynamics, this goal is achieved by taking into account a set of critical exponents, from which we can gain qualitative insights into the critical behavior. These critical exponents are found to be universal to a large extent (only depending on the dimensionality, symmetry etc) and satisfy scaling laws, which can be attributed to scaling hypothesis. Critical behavior of black holes accompanied with their critical exponents have been investigated not only in asymptotically flat space time [42]- [48] but also in the de Sitter and anti de Sitter space [49]- [55].
In this paper, we would like to focus our attention on the critical behavior and geometrothermodynamics of static and spherically symmetric black holes with conformal anomaly. As we know, conformal anomaly, an important concept with a long history, has various applications in quantum field theory in curved spaces, string theory, black hole physics and cosmology. So it is worth probing its influences in phase transitions of black holes. Recently, Cai et al. [56] found a class of static and spherically symmetric black holes with conformal anomaly, whose thermodynamic quantities were also investigated in the same paper. It was found that there exists a logarithmic correction to the well-known Bekenstein-Hawking area entropy. This discovery is quite important in the sense that with this term one is able to compare black hole entropy up to the sub-leading order, in the gravity side and in the microscopic statistical interpretation side [56]. Based on the metrics in that paper, phase transitions of a spherically symmetric Schwarzschild black hole have been investigated by taking into account the back reaction through the conformal anomaly of matter fields recently [57]. It has been shown that there exists an additional phase transition to the conventional Hawking-Page phase transition. The entropy of these black holes has also been investigated by using quantum tunneling approach [58]. Moreover, Ehrenfest equation has been applied to investigate this class of black holes [59] and it has been found that the phase transition is a second order one. Despite of these achievements, there are still many issues left to be explored. Ref. [57] mainly focus on the uncharged case . So it is natural to ask what would happen to the charged black holes. Ref. [59] concentrated their efforts on the Ehrenfest equation in the grand canonical ensemble. So it is worthwhile to study the phase transition in canonical ensemble. The dependence of the phase structure on the parameter deserves to be further investigated. One may also wonder whether the thermodynamic geometry and scaling laws still works to reveal the phase structure and critical behavior when conformal anomaly is taken into consideration. Motivated by these, we would like to investigate the phase transition, geometrothermodynamics and critical exponents in canonical ensemble.
The organization of our paper is as follows. In Sec. 2, the thermodynamics of black holes with conformal anomaly will be briefly reviewed. In Sec. 3, phase transitions will be investigated in details in canonical ensemble and some interesting and novel phase transition phenomena will be disclosed. In Sec. 4, geometrothermodynamics will be established to examine the phase structure we find in Sec. 3. In Sec. 5, critical exponents will be calculated and the scaling laws will be examined. In the end, conclusions will be drawn in Sec. 6.

A brief review of thermodynamics
The static and spherically symmetric black hole solution with conformal anomaly has been proposed as [56] The Newton constant G has been set to one. Both M and Q are integration constants. And the coefficientα is positive. The physical meanings of M and Q were discussed in Ref. [56]. M is nothing but the mass of the solution while Q should be interpreted as U (1) charge of some conformal field theory. When M = Q = 0, the metric above reduces to implying that the vacuum limit is the Minkowski space-time.
In the large r limit, Eq.(2. 2) becomes which behaves like the Reissner-Norström solution.
Solving the equation f (r) = 0, we can get the radius of black hole horizon r + , with which the mass of the black hole can be expressed as The Hawking temperature can be derived as .
The potential difference between the horizon and the infinity can be written as The entropy was reviewed in Ref. [59] as

Novel phase transition phenomena
In this section, we would like to investigate the phase transition of black holes with conformal anomaly in canonical ensemble where the charge of the black hole is fixed. The corresponding specific heat can be calculated as .
Apparently, C Q may diverge when which suggests a possible phase transition. However, the phase transition point characterized by Eq.(3.2) is not apparent. To gain an intuitive understanding, we plot Fig.1(a) using Eq.(3.1). To check whether the phase transition point locates in the physical region, we also plot the Hawking temperature using Eq.(2.7) in Fig.1(b). It is shown that the phase transition point locates in the positive temperature region. From Fig.1(a) and Fig.1(b), we find that there have been striking differences between the caseα = 0 and the caseα = 0. In the case Q = 1,α = 0.1, there are two phase transition point while there is only one in the caseα = 0. The temperature in the caseα = 0 increases monotonically while there exists local minimum temperature in the case Q = 1,α = 0.1. Fig.1(a) can be divided into four phases. The first one is thermodynamically stable (C Q > 0)with small radius. The second one is unstable (C Q < 0)with meidium radius. The third one is thermodynamically stable (C Q > 0)with medium radius while the fourth one is thermodynamically unstable (C Q < 0)with large radius. So the phase transition take place not only from an unstable large black hole to a locally stable medium black hole but also from an unstable medium black hole to a locally stable small black hole. From Fig.1(b),we notice that the Hawking temperature has a local minimum value. And the corresponding location can be derived It is quite interesting to note that the numerator of Eq.(3.3) is the same as Eq.(3.2), which implies that the location which corresponds to the minimum Hawking temperature also witnesses the existence of phase transition.
To probe the dependence of phase transition location on the choice of parameter, we solve Eq.(3.2) and obtain the location of phase transition point as With Eq.(3.4) at hand, we plot Fig.2(a) and Fig.2(b) which show the influence of parameters Q andα respectively. It can be observed from Fig.2(a) and Fig.2(b) that black holes with conformal anomaly have much richer phase structure than that without conformal anomaly. Whenα = 0, the location of the phase transition r c is proportional to the charge Q. However, the cases of black holes with conformal anomaly are quite complicated. Forα = 0.1, the curve can be divided into three regions. Through numerical calculation, we find that black holes have only one phase transition point when Q ⊂ (0, 0.4472). When 0.4472 < Q < 0.7746, there would be no phase transition at all. When Q > 0.7746, there exist two phase transition points, just as what we show in Fig.1(a). And the distance between these two phase transition point becomes larger with the increasing of Q. Fig.2(b) shows the case that the charge Q has been fixed at one. We notice that there would be two phase transition points when 0 <α < 1 6 , which is consistent with Fig.1(a). And the distance between these two phase transition point becomes narrower with the increasing ofα. Whenα ⊂ ( Apart from the specific heat, we would also like to investigate the behavior of the inverse of the isothermal compressibility, which is defined as Utilizing the thermodynamic identity relation where the second term on the right hand side can be calculated through .

(3.9)
We show the behavior of κ −1 T in Fig.4. Comparing Fig.4 with Fig.1(a), we find that the inverse of the isothermal compressibility κ −1 T also diverges at the critical point.

Geometrothermodynamics
According to geometrothermodynamics [31], the (2n + 1)-dimensional thermodynamic phase space T can be coordinated by the set of independent quantities {φ, E a , I a }, where φ corresponds to the thermodynamic potential, and E a , I a are the extensive and intensive As proposed by Quevedo, the non-degenerate metric G and the thermodynamic metric g can be written as follows [38] where η ab = diag(−1, · · · , 1). To construct geometrothermodynamics of black holes with conformal anomaly in canonical ensemble, we choose M to be the thermodynamic potential and S, Q to be the extensive variables. Then the corresponding thermodynamic phase space is a 5-dimensional one coordinated by the set of independent coordinates{M, S, Q, T, Φ}. The fundamental Gibbs 1-form defined on T can then be written as The non-degenerate metric G from Eq.(4.1) can be changed into the space of thermodynamic equilibrium states can be induced. According to Eq.(4.1), the thermodynamic metric g can be written as follows Utilizing Eqs.(2.6) and (2.9), we can easily calculate the relevant quantities in Eq.(4.6) as × [r 4 + + 2r 2 +α + Q 2 (3r 2 + − 16α) + 8α(Q 2 − r 2 + − 2α)lnr + ], (4.12) .

(4.13)
Utilizing Eqs.(4.12) and (4.13), we can obtain the Legendre invariant scalar curvature as where The numerator of the Legendre invariant scalar curvature is too lengthy to be displayed here. From Eq.(4.15), we find that the Legendre invariant scalar curvature shares the same factor r 4 + + 10r 2 +α − 8α 2 + Q 2 (4α − 3r 2 + ) with the specific heat C Q in its denominator , which implies that it would diverge when r 4 + + 10r 2 +α − 8α 2 + Q 2 (4α − 3r 2 + ) = 0. That is the exact condition that the phase transition point satisfies. To get an intuitive sense on this issue, we plot Fig.5 showing the behavior of thermodynamic scalar curvature R Q . From Fig.5, we find that thermodynamic scalar curvature R Q diverges at three locations. Comparing Fig.5 with Fig.1(b), we find that the second diverging point which corresponds to negative Hawking temperature does not have physical meaning. Furthermore, the first and the third diverging points coincide exactly with the phase transition point, which can be witnessed by comparing Fig.5 with Fig.1(a). So we can safely draw the conclusion that the Legendre invariant metric constructed in geometrothermodynamics correctly produces the behavior of the thermodynamic interaction and phase transition structure of black holes with conformal anomaly.

Critical exponents and scaling laws
In order to have a better understanding of the phase transition of black holes with conformal anomaly, we would like to investigate their critical behavior near the critical point by considering a set of critical exponents in this section. Before embarking on calculating critical exponents, we would like to reexpress physical quantities near the critical point as where |∆| ≪ 1, |ε| ≪ 1, |η| ≪ 1. Note that the footnote "c" in this section denotes the value of the physical quantity (or the expression) at the critical point. For example, T c corresponds to the temperature at the critical point. Critical exponent α is defined through To obtain T − T c , we would like to carry out Taylor expansion as below ) Q=Qc ] r + =rc (r + −r c ) 2 +higher order terms, In the above derivation, we have considered the fact that C Q diverges at the critical point, making the second term at the right hand side of Eq.(5.5) vanish. Substituting Eq.(5.1) into Eq.(3.1) and keeping only the linear terms in its denominator, we obtain which can be transformed via Eq.(5.6) into Comparing Eq.(5.9) with Eq.(5.4), we can obtain α = 1/2. Critical exponent β is defined through the following relation when Q is fixed, The above definition motivates us to carry out the Taylor expansion as Φ(r + ) = Φ c + [( ∂Φ ∂r + ) Q=Qc ] r + =rc (r + − r c ) + higher order terms, (5.11) Utilizing Eq.(2.8), (5.11) and neglecting higher order terms of Eq.(5.10), we get Comparing Eq.(5.10) with Eq.(5.12), we can obtain β = 1/2. Critical exponent γ is defined through the following relation Substituting Eq.(5.1) and (5.6) into Eq.(3.9) and keeping only the linear term of ∆, we obtain (5.14) From Eq.(5.13) and (5.14), we find that γ = 1/2 Critical exponent δ is defined for the fixed temperature T c through To obtain Q − Q c , we would like to carry out Taylor expansion as ) T ] r + =rc (r + − r c ) 2 + higher order terms, (5.16) Utilizing the thermodynamic identity again, we get In the above derivation, we have taken into account the fact that C Q diverges at the critical point, making the first term at the right hand side of Eq. where Taylor expanding Φ near the critical point, we get Φ(r + ) = Φ c + [( ∂Φ ∂r + ) T ] r + =rc (r + − r c ) + higher order terms, (5.20) where the coefficient of the second term on the right hand side can be derived as follows Utilizing Eq.(5.18), (5.20), (5.21),we get from which we can draw the conclusion that δ = 2. Critical exponent ϕ is defined through from which we obtain ψ = 1/2. Till now, we have finished the calculations of six critical exponents. They are also equal to 1/2 except δ = 2. Our results are in accordance with those in classical thermodynamics. And it can be easily proved that critical exponents we obtain in our paper satisfy the following thermodynamic scaling laws (5.28)

Conclusions
The phase transition of black holes with conformal anomaly has been investigated in canonical ensemble. Firstly, we calculate the relevant thermodynamic quantities and discuss the behavior of the specific heat at constant charge. We find that there have been striking differences between black holes with conformal anomaly and those without conformal anomaly. In the case Q = 1,α = 0.1, there are two phase transition point while there is only one in the caseα = 0. The temperature in the caseα = 0 increases monotonically while there exists local minimum temperature in the case Q = 1,α = 0.1. This local minimum temperature corresponds to the phase transition point. We also find that the phase transitions of black holes with conformal anomaly take place not only from an unstable large black hole to a locally stable medium black hole but also from an unstable medium black hole to a locally stable small black hole. We also study the behavior of the inverse of the isothermal compressibility κ −1 T and find that κ −1 T also diverges at the critical point. Secondly, we probe the dependence of phase transitions on the choice of parameters. The results show that black holes with conformal anomaly have much richer phase structure than that without conformal anomaly. Whenα = 0, the location of the phase transition r c is proportional to the charge Q. By contrast, the case of black holes with conformal anomaly is more complicated. Forα = 0.1, the curve can be divided into three regions. Through numerical calculation, we find that black holes has only one phase transition point when Q ⊂ (0, 0.4472). When 0.4472 < Q < 0.7746, there would be no phase transition at all. When Q > 0.7746, there exist two phase transition points. And the distance between these two phase transition points becomes larger with the increasing of Q. In the case that the charge Q has been fixed at one, we notice that there would be two phase transition point when 0 <α < 1 6 . And the distance between these two phase transition points becomes narrower with the increasing ofα. Whenα ⊂ ( 1 6 , 1 2 ), there would be no phase transition. Whenα > 1 2 , there would be only one phase transition point. Thirdly, we build up geometrothermodynamics in canonical ensembles. We choose M to be the thermodynamic potential and build up both thermodynamic phase space and the space of thermodynamic equilibrium states. We also calculate the Legendre invariant thermodynamic scalar curvature and depict its behavior graphically. It is shown that Legendre invariant thermodynamic scalar curvature diverges exactly where the specific heat diverges. Based on this, we can safely conclude that the Legendre invariant metrics constructed in geometrothermodynamics can correctly produce the behavior of the thermodynamic interaction and phase transition structure even when conformal anomaly is taken into account.
Furthermore, we calculate the relevant critical exponents. They are also equal to 1/2 except δ = 2. Our results are in accordance with those of other black holes. And it has been proved that critical exponents we obtain in our paper satisfy the thermodynamic scaling laws. We conclude that the critical exponents and the scaling laws do not change even when we consider conformal anomaly. This may be attributed to the mean field theory.