Parametric phase transition for Gauss-Bonnet AdS black hole

With the help of the parametric solution of the Maxwell equal area law for the Gauss-Bonnet AdS black hole in five dimensions, we find the second analytical solution to the first order phase transition. We analyze the asymptotic behaviors of some characteristic thermodynamic properties for the small and large black holes at the critical and zero temperatures and also calculate the critical exponents and the corresponding critical amplitudes in detail. Moreover, we give the general form of the thermodynamic scalar curvature based on the Ruppeiner geometry and point out that the attractive interaction dominates in both the small and large black hole phases when the first order phase transition occurs in the five dimensional Gauss-Bonnet AdS black hole.


Motivation
The black hole thermodynamics, which stems largely from the pioneering works by Hawking and Bekenstein [1][2][3], has been increasingly considered as the most feasible topic of semiclassical quantum gravity effect. Especially in the paradigm of extended phase space, 1 the black holes in the AdS spacetime can be made an analogy with the van der Waals fluid and such an analogy can show rich thermodynamic critical phenomena [4][5][6][7][8][9][10][11][12]. By using the ordinary thermodynamics, one can handle [13][14][15][16] mainly the first order phase transition of black holes. Unfortunately, it is usually difficult to find an exact and analytical solution to the first order phase transition. Hence, the parametric solution provides [17,18] a new possibility, which not only presents the independent variable governing the first order phase transition, but also shows more detailed information of the phase transition. Our current work aims to study the critical behaviors of the Gauss-Bonnet AdS black hole in five dimensions from the viewpoint of parametric solutions.
We summarize the basic formulas of the Gauss-Bonnet AdS black hole for our later use. The action of Einstein-Gauss-Bonnet gravity theory with a negative cosmological constant Λ 1 Here it means that the cosmological constant and its conjugate are treated as the thermodynamic pressure and volume variables, respectively.
In the present paper we will investigate the phase transition and critical behaviors for the Gauss-Bonnet AdS black hole in d = 5 dimensions. Our main motivations are listed as follows.
• Hitherto, a complete quantum gravity theory has not been established yet. Whatever the quantum gravity theory would be, higher order corrections to the usual Einstein-Hilbert action should exist. The Gauss-Bonnet theory of higher order curvature corrections is an expected candidate. On the other hand, the Gauss-Bonnet theory is a special case of the Lovelock gravity theory. So it is a general second order covariant gravity theory in dimensions higher than four [34]. According to the explanation of the Gauss-Bonnet coefficient (see footnote 2), the Gauss-Bonnet terms can be regarded as corrections from the heterotic string theory [35]. Moreover, the Gauss-Bonnet terms represent from the holographical viewpoint part of the 1/N correction to the large N limit of the dual SU(N)-like gauge field theory [36]. Therefore, the investigation of the Gauss-Bonnet AdS black hole is interesting and necessary in its own right.
• For the Gauss-Bonnet AdS black hole, the spacetime dimension is d ≥ 5. 3 In addition, based on the discussion about thermodynamic behaviors of the Gauss-Bonnet AdS black hole [19,20,33], one can observe that its thermodynamic behavior in d = 5 is different from that in other dimensions. Hence, the dimension d = 5 is of special importance. 2 The Lovelock coefficient is proportional to the inverse string tension in string theory [21]. Hence, we take α GB > 0 in this paper for Gauss-Bonnet AdS black holes. 3 The Gauss-Bonnet term is just a topological term and the Gauss-Bonnet AdS black hole degenerates into a normal AdS black hole in the case of d = 4.
• Based on the thermodynamics of black holes in the extended phase space, black holes undergo one first order phase transition which is analogous to that of van der Waals fluid [10]. The Maxwell equal-area law is a very useful tool to deal with the first order phase transition [13,16]. Unfortunately, it is very difficult to obtain the exact analytical solution of the Maxwell equal-area law. For black holes, the only known exact analytical solution [13] at present is about RN-AdS black hole in d = 4 dimensions. 4 In addition, the corresponding parametric analytical solution is also known [18]. For other types of black holes, the numerical methods are often used to analyze phase transitions. Fortunately, we work out the parametric analytical solution of the Maxwell equal-area law for the Gauss-Bonnet AdS black hole in d = 5 dimensions, which may be thought of as the second analytical solution.
• With the help of the parametric analytical solution of the first order phase transition, we can analyze in detail the critical behaviors at the critical and zero temperatures for the Gauss-Bonnet AdS black hole in d = 5 dimensions. At the same time, we can also calculate directly the critical exponents and the corresponding critical amplitudes.
The paper is organized as follows. In section 2, we briefly review some basic thermodynamic properties and the formula of the Maxwell equal area for the Gauss-Bonnet AdS black hole in d = 5 dimensions. In section 3, we give the parametric analytical solution of the Maxwell equal-area law for the Gauss-Bonnet AdS black hole in d = 5 dimensions. We then use the parametric solution to analyze in detail the critical behaviors at the critical and zero temperatures in section 4. Finally, we devote to drawing our conclusion in section 5. In addition, the general form of the thermodynamic scalar curvature based on the Ruppeiner geometry is attached in Appendix A. This scalar curvature is calculated for the study of interaction in the small and large black hole phases from the viewpoint of Ruppeiner thermodynamic geometry. Throughout this paper, we adopt the units = c = k B = G = 1.
2 Thermodynamics for Gauss-Bonnet AdS black hole in d = 5 The basic thermodynamic properties of the Gauss-Bonnet AdS black hole take the following forms in terms of the horizon radius r h which is determined by the zero point of g −1 rr component in the metric [27][28][29][30][31][32][33], (2.5) From the equation of state eq. (2.5), one can obtain [27][28][29][30][31][32][33] the critical values of the Gauss-Bonnet AdS black hole in d = 5 dimensions, where the inflection point in the P − V graph determined by has been utilized. Below these critical values, there is an oscillating part in the P − V graph. In other words, there exists three black hole phases with different sizes at the same pressure when T < T c . The three sizes of black holes are marked by large, small, and medium-sized black holes, respectively. The medium-sized black hole corresponds to an oscillating part in the P − V graph and is unstable due to its negative heat capacity at constant pressure. Hence we have to utilize the Maxwell equal area law to replace this oscillating part by an isobar, which implies that the small black hole can jump to the large one through the isobaric process. This phenomenon is called the first order phase transition. It satisfies [33] the following equations, where P stands for an isobar, and V 1 and V 2 denote the (thermo)volume defined by eq. (2.4) for the small and large black holes with the horizon radii r 1 and r 2 , respectively. In the next section we will parameterize the solution of the equation eq. (2.8) and analyze its relevant critical behaviors.

Parametric phase transition
Substituting eq. (2.5) into eq. (2.8), we obtain a key expression 5 for parametric phase transition, It is the result that allows us to parameterize the phase transition for the Gauss-Bonnet AdS black hole. The specific process is as follows. Based on eq. (3.1), it is crucial to introduce a dimensionless auxiliary parameter y that parameterize [33] the horizon radii of the small and large black holes in the following way, Some dimensionless reduced parameters are usually introduced for the sake of convenience, wheren is the dimensionless reduced number density, 6 and 0 ≤t ≤ 1 and 0 ≤p ≤ 1. Inserting eq. (3.2) into eq. (2.8) and considering eq. (3.3), we derive the dimensionless reduced temperature and pressure of parametric first order phase transition for the Gauss-Bonnet AdS black hole,t = 3(y 3 + y) y 4 + 4y 2 + 1 ,p = 6y 2 y 4 + 4y 2 + 1 .
In this way, we can give a relationship between the thermodynamic situations and the limits of the auxiliary parameter y. Because of r 2 ≥ r 1 , we see 0 ≤ y ≤ 1. More precisely, we can obtain For the small black hole phase, we havê

7)
5 This expression first appeared in ref. [33], where it was used to numerically analyze the phase transition and latent heat. 6 Here N is the total number of molecules in the black hole system, and the number density is defined by n ≡ N/V . In addition, the critical number density is n c = N/V c , so we obtain the dimensionless reduced number densityn := n/n c = 1/v. and for the large black hole phase, we havê At the same time, due to eq. (3.4) we can also get the slope of the co-existence curve of the small and large black hole phases, dp dt = 4(y 3 + y) y 4 + 1 . (3.9) Furthermore, when the first order phase transition occurs, i.e. the small black hole phase jumps to the large one, we can give the difference in volume, entropy and number density between the small and large black hole phases at the two edges of the co-existence curve, and the latent heat L of the first order phase transition on crossing the small-large black hole phase co-existence curve, L P c V c = 12(1 + y 2 ) 2 (1 − y 4 ) y 2 (y 4 + 4y 2 + 1) , (3.11) where the Clausius-Clapeyron equation has been used. 7 Finally, we compute the reduced thermodynamic scalar curvatures 8 of the small and large black hole phases for the Gauss-Bonnet AdS black hole in d = 5 dimensions, 16(y 4 + 4y 2 + 1) 3y(y 2 + 1)(3y 2 + 1) 2 , (3.12a) (3.12b) 7 Considering the Clausius-Clapeyron equation [37], and the dimensionless reduced parameters in eq. (3.3), we can write the latent heat, Again using eq. (2.6), we have the relation: L PcVc = 8L TcSc . 8 See Appendix A for the details. Figure 1 shows the co-existence curve of small and large black hole phases (described by eq. (3.4)), the latent heat on crossing the small-large black hole phase co-existence curve (described by eq. (3.11)), and the number densities and the thermodynamic scalar curvatures of small and large black hole phases with respect to the temperature, respectively. The coexistence curve begins at the origin and terminates at the critical point. Above the critical point, we cannot distinguish the two phases of small and large black holes. The latent heat goes to infinity with the temperature tending to zero and it approaches zero with the temperature tending to the critical value. This is consistent with the result that was numerically calculated in ref. [33]. For the large black hole, its number density increases with the temperature increasing. However, for the small black hole, its number density decreases with the temperature increasing and it tends to infinity when the temperature goes to zero. Both the thermodynamic scalar curvatures of small and large black holes are negative, which implies that the attractive interaction dominates. Meanwhile, we notice that the thermodynamic scalar curvature of the small black hole will go to negative infinity with the temperature tending to zero. As to the case at the critical or zero temperature, the critical behaviors of the small and large black hole phases will be discussed in the next section.

Parametric critical behavior 4.1 Approaching the critical temperature
In order to investigate more clearly the thermodynamic behaviors of the small and large black hole phases at the critical temperature, we define the following two parameters, where they satisfy the inequalities: 0 ≤ τ ≤ 1 and 0 ≤ x ≤ 1. From eq. (3.5), we see that τ → 0 and x → 0 in the critical situation. Substituting eq. (4.1) into the reduced temperature eq. (3.4) and making the Taylor expansion at x = 0, we obtain Then we solve x reversely from the above relation,

Small black hole at the critical temperature
Substituting eq. (4.1) into eq. (3.7) and making the Taylor expansion at x = 0, and considering eq. (4.3) again, we obtain (4.4c) and the reduced thermodynamic scalar curvature,

Large black hole at the critical temperature
Making a similar discussion to that in the above subsection except for substituting eq. (4.1) into eq. (3.8), we eventually havê (4.6c) and the reduced thermodynamic scalar curvature, Here we make two comments to eqs. (4.4), (4.5), (4.6), and (4.7). The first is that the thermodynamic behaviors of the small and large black hole phase are symmetrically distributed on both sides of the critical point. The other comment is that the leading order term related to τ 1/2 in each of the equations will play a major role in calculating the static critical exponents in the next subsection.

Static critical exponent
Critical exponents describe the behavior of thermodynamic quantities when the temperature approaching phase transition points and they do not depend on the details of physical systems, i.e., they are universal [10,37]. For the sake of calculation of critical exponents, one usually introduces the following notations, • α and α ′ are related to the specific heat at constant volume: where a and a ′ are the critical amplitudes that correspond to the critical exponents α and α ′ , respecttively. For the Gauss-Bonnet AdS black hole, due to C V = 0, it is easy to have α = α ′ = 0, a = a ′ = 0. (4.10) • β is related to the difference between the small and large black hole number densities on crossing the co-existence curve: Note that one cannot define a critical exponent and its amplitude for the patht → 1 + since the order parameter ∆n is zero fort > 1. According to eqs. (4.4c) and (4.6c), we obtain ∆n = 8 √ 6τ 1/2 . So the critical exponent and amplitude are • γ is related to the isothermal compressibility: where κ T is isothermal compression coefficient. By means of inserting eqs. (4.14) Thus we directly read the critical exponent and its amplitude, In addition, for the patht → 1 − , substituting eqs. (4.1) and (4.8) into the reduced pressure eq. (3.4) and making the Taylor expansion at x = 0, we obtain p 0 = −4τ with the help of eq. (4.3). Next, considering eqs. (4.4a) and (4.6a), we have v 2 0 = (4 √ 6) 2 τ . Combing the relations of p 0 and v 2 0 , we deduce p 0 = −v 2 0 /24. As a result, we derive the desired critical behavior, from which the critical exponent and amplitude can be read • δ is related to the critical behavior in the critical isothermt = 1: Here we list the critical exponents and critical amplitudes about the van der Waals fluid and Gauss-Bonnet AdS black hole in d = 5 dimensions in Table 1. One can see that the critical exponents are same for the two systems, which implies that the critical exponents are universal. However, the critical amplitudes of the two systems are different from each other, which shows that the critical amplitudes present characteristics of systems, i.e. they can be used to distinguish different thermodynamic systems.

Exponent
Definition van der Waals fluid Gauss-Bonnet AdS BH Exponent Amplitude Exponent Amplitude

Approaching zero temperature
The thermodynamic behaviors of the small and large black hole phases at zero temperature for the Gauss-Bonnet AdS black hole in d = 5 dimensions will be discussed in detail in this subsection. From eq. (3.6), we see that for studying the critical behaviors at zero temperature we need to make the Taylor expansion for the reduced temperature eq. (3.4) at y = 0, t = 3y − 9y 3 + 33y 5 + O(y 7 ), (4.20) and then we solve y reversely from the above relation, The reduced number densities: The reduced thermodynamic scalar curvatures: . (4.25b) From the above formulas, we can clearly see that when the temperature approaches zero, the critical values of the number density and thermodynamic scalar curvature indeed go to infinity for the small black hole, but the two values go to zero for the large black hole. The reason is that the (thermo)volume or entropy of small (large) black hole goes to zero (infinity) when the temperature approaches zero. The analytic analysis is consistent with the description of Figure 1.

Summary
Based on the parametric analytical solution of the first order phase transition for the Gauss-Bonnet AdS black hole in d = 5 dimensions, we investigate the thermodynamic critical behaviors. This parametric solution may be regarded as the second analytic solution of the Maxwell equal-area law. At the critical temperature, we have calculated the critical exponents and the corresponding critical amplitudes in detail. By comparing them with that of the van der Waals fluid, we can see clearly that the critical exponents are universal, but the critical amplitudes present characteristics of thermodynamic systems. By means of the thermodynamic scalar curvature, we have obtained that the attractive interaction dominates in both the small and large black hole phases. In addition, we have also analyzed the asymptotic behaviors of thermodynamic properties for the small and large black holes at zero temperature. With the help of the parametric solution of the Maxwell equal-area law, we have acquired more details about the first order phase transition of black holes. Our treatment can be extended to other black hole models and the key point is to select such an appropriate auxiliary parameter that an analytical solution can be found.

Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under grant No.11675081.

A Thermodynamic scalar curvature
The Ruppeiner geometry is a powerful tool to investigate microscopic properties of black holes completely from the thermodynamic point of view [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. Its metric can be written in the Weinhold energy form [53], where X α = (S, P ). The first law of black hole thermodynamics takes the following form, dM = T dS + V dP + · · · . (A.2) Note that both the enthalpy M and temperature T are linearly dependent on pressure P , which implies ∂ 2 M/∂P 2 = 0 and ∂ 2 T /∂P 2 = 0. Based on the facts, one can obtain the general form of the thermodynamic scalar curvature induced by eq. (A.1), Importantly, the thermodynamic scalar curvature can qualitatively reflect some information of the internal interaction of a thermodynamic system. That is, a positive (negative) thermodynamic scalar curvature implies a repulsive (attractive) interaction, while a vanishing thermodynamic scalar curvature implies no interaction [40,41].
Naturally, we compute the thermodynamic scalar curvature for the Gauss-Bonnet AdS black hole in d = 5 dimensions with the help of eqs.