Triple points and phase diagrams of Born-Infeld AdS black holes in 4D Einstein-Gauss-Bonnet gravity

By treating the cosmological constant as a thermodynamic pressure, we investigate the thermodynamic behaviors of Born-Infeld AdS black hole in 4D Einstein-Gauss-Bonnet (EGB) gravity. The result shows that the Van der Waals like small/large black hole (SBH/LBH) phase transition always appears for any positive parameters $\alpha$ and $\beta$. Moreover, we observe a new phenomenon of small/intermediate/large black hole (SBH/IBH/LBH) phase transition with one tricritical and two critical points in the available parameter region for \alpha and \beta. This behavior is reminiscent of the solid/liquid/gas phase transition.

tion gravities, the most extensively studied theory is the so-called Gauss-Bonnet(GB) gravity [24].
However, in four dimensions the GB term is a total derivative, and has no influence on the field equation. It's interesting to note that some authors considered the GB gravity by rescaling the GB coupling parameter α → α/(D − 4) in four dimensions [25,26], and then the GB term gives rise to non-trivial dynamics. Taking the similar technique, Glavan and Lin [27] presented 4D Einstein Gauss-Bonnet gravity(4D EGB). Its main feature is that the static spherically symmetric solution of this theory is free from the singularity problem for a positive GB coupling constant. Later, the solutions of Maxwell charged GB AdS black hole [28], Hayward AdS [29] and Bardeen AdS [30] black holes have been also constructed in the 4D EGB gravity. Moreover, the thermodynamics and phase transition of Maxwell charged GB AdS black hole [31][32][33][34], nonlinear electrodynamics charged GB AdS black hole [35] and Bardeen AdS [36] black hole further have been also discussed in the extended phase space.
As well known, a point-like charge in Maxwell's electromagnetic field theory usually brings about infinite self-energy, since it is allowed a singularity at the charge position. Then, Born, Infeld [37] and Hoffmann [38] introduced Born-Infeld(BI) electromagnetic field to overcome infinite selfenergy problem by imposing a maximum strength of the electromagnetic field. Moreover, BI type effective action arises in an open superstring theory and D-branes are free of physical singularities.
In recent years, the solution and related thermodynamic properties of BI black hole have received some attentions [39,40]. Particularly, in the extended space, the BI AdS black hole exhibits an interesting reentrant phase transition in four dimensional Einstein-Born-Infeld(EBI) gravity, besides the usual Van der Waals liquid-gas like SBH/LBH phase transition [9,41]. Recently, Ref. [42] proposed a static and spherically symmetric BI AdS black hole solution in the novel 4D EGB gravity, and analyzed some basic thermodynamics of the BI AdS black hole. In this paper, we will generalize the discussions to consider the possible phase transition and critical phenomena for BI AdS black holes in the extended phase space. This paper is organized as follows. In Sec. II, we reconsider the BI AdS black hole solution and its thermodynamics in the 4D EGB gravity. In Sec. III, we will investigate phase transition and critical behavior of BI AdS black holes in the 4D EGB gravity. We will summarize our results in Sec. IV.
The action of D-dimensional EGB gravity minimally coupled to BI electrodynamics field in the presence of a negative cosmological constant is given by [42] is a negative cosmological constant, G is the Gauss-Bonnet term R 2 − 4R µν R µν + R µνρσ R µνρσ , α is the Gauss-Bonnet coefficient, and L BI is the Lagrangian of the BI electrodynamics with In the low energy effective action of heterotic string theory, α is proportional to the inverse string tension with positive coefficient [25]. Thus in this paper we focus on the case with a positive Gauss-Bonnet coefficient α > 0.
In the limit D → 4, the spherically symmetric solution takes the following form [42] where M and Q are the mass and charge of BI AdS black hole respectively, and 2 F 1 is the hypergeometric function. The electromagnetic potential difference (Φ) between the horizon and infinity In the limit of β → ∞, f (r) recovers the Maxwell charged RN-AdS-like black hole solution in the 4D EGB gravity [28] f (r) = 1 + r 2 When the GB term is switched off, i.e., α = 0, f (r) reduces to the BI AdS black hole solution obtained in the Einstein-Born-Infeld(EBI) gravity [39,40].
In terms of the horizon radius r + , the mass M , Hawking temperature T and entropy S of four dimensional BI AdS black hole can be written as In the extended phase space, the cosmological constant Λ is regarded as a variable and also identified with the thermodynamic pressure P = − Λ 8π in the geometric units G N = = c = k = 1. Then, the black hole mass M is considered as the enthalpy H rather than the internal energy of the gravitational system. The corresponding thermodynamical volume V is given by With all the above thermodynamic quantities at hand, it is easily verified that the first law of black where B is BI vacuum polarization and A is the conjugate quantity of GB coupling parameter α [42].
The behavior of Gibbs free energy G is important to determine the thermodynamic phase transition. The free energy G obeys the following thermodynamic relation G = H − T S with

III. PHASE TRANSITION OF BORN-INFELD ADS BLACK HOLE
From the Hawking temperature (7), we can obtain the equation of state The crucial information about the equation of state is encoded in the number of critical points it admits. As usual, a critical point occurs when P has an inflection point Then we can obtain corresponding critical values where r c is determined by Here P c , r c and T c are all positive in order the critical point to be physical.
Now we consider the critical behaviors of BI AdS black hole in the extended phase space.
Unfortunately, an analytic solution is not possible, since higher-order polynomials for Eq. (16) are encountered. Then, we proceed numerically: for a given α and β, we solve the Eqs. (14)(15)(16) for critical r c , P c and T c , and then calculate the values of P , T and G using Eqs. (7) and (11), yielding a G − T diagram. Once the behavior of G is known, we can further compute the corresponding phase diagram, coexistence lines, and critical points in the P − T plane.
In the limit of β → ∞, Eq.(3) reduces to that of charged AdS black hole [28], where it recovered the existence of Van der Waals like SBH/LBH phase transition in the extended phase space [31].
On the other hand, in the limit of α → 0, Eq. (3)  and the disappearance of phase transition if β < 1 √ 8Q ≈ 0.35355 Q . Inspiring by these rich phase transition structures for BI AdS black hole in the EBI gravity, here we firstly consider the small contributions of Gauss-Bonnet term (i.e., α = 0.01) to BI AdS black hole in the 4D EGB gravity. Moreover, we assume Q = 1 for simplification in future. Then we find that there always exists one physical critical point when β ∈ (0, 0.339], even though the BI coupling parameter β is less than 1 √ 8 . Taking β = 0.26(< 1 √ 8 ) for example, we obtain the critical horizon r c = 0.4459, critical temperature T c = 0.1042 and critical pressure P c = 0.0374. The P − r + isotherm diagram is plotted in Fig.1(a). The dotted line corresponds to the "idea gas" phase behavior when T > T c , and the Van der Waals like SBH/LBH phase transition appears in the system when T < T c . The behavior of the Gibbs free energy G is important to determine the thermodynamic phase transition.
We see that the G surface in Fig.1(b) demonstrates the characteristic "swallow tail" behavior, which shows the Van der Waals like SBH/LBH phase transition in the system for T < T c . The coexistence line in the (P, T ) plane by finding a curve where the Gibbs free energy and temperature coincide for small and large black holes. The coexistence line is very similar to that in the Van der Waals fluid. The critical point is shown by a small circle at the end of the coexistence line, see Fig.1(c).  Fig.2(a) with different temperature T =0.0435, 0.0453 and 0.0404 (three critical temperatures). In particular, there exists a special isotherm curve (T = T τ ), which denotes two Van der Waals oscillations such that the two equal area laws saturate for the same tricritical pressure P τ , see Fig.2(b). The corresponding Gibbs free energy is illustrated in Fig.3. When P > P c1 , there is no such behavior, and no phase transition occurs. In the ranges P c2 < P < P c1 , it displays one characteristic swallow tail behavior in Fig.3(b). On the other hand, the situation becomes more subtle in the range P c3 < P < P c2 , see Figs.3(c)-3(e). For fixed P satisfying P τ < P < P c2 , we observe two first-order small/intermediate and intermediate  The corresponding P − T diagram is displayed in Fig. 4. It is clear that there is a triple point, at which the small, large and intermediate black holes can coexist together. In the region of P τ and P c2 , there will be a SBH/IBH/LBH phase transition, which ends at P c2 . In the mean time, the first order SBH/IBH phase transition emerges from P τ and terminates at critical P c1 . In addition, the system below this pressure P τ will undergo a Van der Waals like SBH/LBH phase transition of first order with the decreasing of T . Notice that the critical point (P c3 , T c3 ) is absent from this phase diagram, since the first critical point measures the appearance of the second swallow tail, which does not participate in phase transition. This available parameter region for parameters α and β is plotted in Fig.6, where the vertical axis denotes the EBI AdS black hole with α → 0. It's worth to note that the Refs. [9,41] show the existence of an interesting reentrant phase transitions (RPT), besides the usual SBH/LBH phase transitions are observed in the region of 1 √ 8 ≤ β ≤ 1 2 , since the EBI AdS black hole possesses three real roots r c , but only two physical positive roots r c . Therefore, we have reason to believe that for the BI AdS black hole in 4D Gauss-Bonnet gravity, the Gauss-Bonnet term results in three real positive roots r c1 , r c2 and r c3 of Eq. (16), and corresponding physical positive temperature T c1 , T c2 , T c3 and pressure P c1 , P c2 , P c3 . Moreover, the allowed yellow area for BI parameter β becomes smaller with the increasing of α, and finally disappears when α c = 0.04601 and β c = 0.28935.

IV. DISCUSSION AND CONCLUSION
In this paper, we discuss the thermodynamic behaviors of Born-Infeld AdS black hole in the novel 4D EGB gravity by treating the cosmological constant in an extended phase space. We have written out the equations of state and examined the phase structures with positive parameter α of Gauss-Bonnet term and parameter β of Born-Infeld term. We found there exist various thermodynamic phenomena, such as Van der Waals behaviour and tripe points. Taking α = 0.01 for example, the system exhibits a first order SBH/LBH phase transition which resembles the Van der Waals liquid-gas phase transition in fluids in the regions of β ∈ (0, 0.339] and β ≥ 0.4. For the region of β ∈ [0.34, 0.4), the system admits a standard SBH/IBH/LBH phase transition, which is reminiscent of a solid/liquid/gas phase transition.
We further investigate the general case for parameters α and β. We found that the triple point does not always exist in that range. The triple point and the SBH/IBH/LBH phase transition are limited in the yellow area. Moreover, the allowed area for BI parameter β becomes smaller with the increasing of α, and finally disappears when α c = 0.04601 and β c = 0.28935. This work is supported by the National Natural Science Foundation of China under Grant Nos.11605152, 11675139 and 51575420, and Outstanding young teacher programme from Yangzhou