Hawking-Page phase transitions of the charged AdS black holes surrounded by quintessence

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Da-Wei Yan, Ze-Rong Huang and Nan Li. Hawking-Page phase transitions of the charged AdS black holes surrounded by quintessence[J]. Chinese Physics C.
Da-Wei Yan, Ze-Rong Huang and Nan Li. Hawking-Page phase transitions of the charged AdS black holes surrounded by quintessence[J]. Chinese Physics C. shu
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Hawking-Page phase transitions of the charged AdS black holes surrounded by quintessence

    Corresponding author: Nan Li, linan@mail.neu.edu.cn
  • 1. Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China
  • 2. Department of Material Physics, School of Material Sciences and Engineering, Northeastern University, Shenyang 110819, China

Abstract: The Hawking-Page phase transitions between the thermal anti-de Sitter vacuum and the charged black holes surrounded by quintessence are studied in the extended phase space. The quintessence field, with a state parameter $-1 < w < -1/3$, modifies the temperature and the Gibbs free energy of the black hole. The phase transition temperature $T_{\rm{HP}}$ and the Gibbs free energy $G$ are first analytically investigated for the special case with $w=-2/3$, and then numerically illustrated for the cases with general $w$. The phase transition temperature $T_{\rm{HP}}$ is found to increase with pressure and to decrease with electric potential. Moreover, $T_{\rm{HP}}$ is also significantly decreased by the quintessence field, which presents a negative pressure around the black hole.

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    Ⅰ.   INTRODUCTION
    • Black hole physics has been one of the most profound topics in physics during the last half century, in which black hole thermodynamics possesses special and fundamental importance, as it shows that a black hole is not merely a mathematical singularity, but should be considered as a physical system with temperature and entropy [1]. In this way, black hole thermodynamics synthesizes the theories of thermodynamics, classical gravity, and quantum mechanics, and thus paves a way to our final understanding of quantum gravity [2].

      In recent years, black hole thermodynamics in the so-called "extended phase space" has received increasing research interests [3]. In contrast to traditional thermodynamics, there is no usual $ p $$ V $ term in the first law of black hole thermodynamics. To restore this $ p $$ V $ term, the black hole phase space must be extended, so as to accommodate effective thermodynamic pressure and volume, and such theory thus bears its present name. In this framework, black hole thermodynamics is studied in the anti-de Sitter (AdS) space, with the cosmological constant $ \Lambda $ being negative. Moreover, if $ \Lambda $ is allowed to change, it plays the role of a positive varying pressure [4],

      $ p = - \frac{\Lambda}{8\pi} = \frac{3}{8\pi l^2}, $

      (1)

      where $ l $ is the AdS curvature radius. Furthermore, an effective thermodynamic volume $ V $ of the black hole can be introduced as the conjugate variable of $ p $. By this means, the missing $ p $$ V $ term reappears in the first law of black hole thermodynamics. However, it is found not to be the usual work term $ -p\, {{\rm{d}}} V $, but $ V\, {{\rm{d}}} p $. Hence, the black hole mass should be identified as its enthalpy instead of internal energy, and this is the distinct character of the black hole thermodynamics in the extended phase space.

      In the extended phase space, an AdS black hole shows great similarities to a non-ideal fluid [3]. For example, for the charged Reissner–Nordström–AdS (RN–AdS) black hole, there exists a large–small black hole transition, extremely analogous to the gas–liquid phase transition of the van der Waals fluid. Also, the critical exponents and equations of corresponding states are exactly the same for these two seemingly unrelated systems. These remarkable observations have aroused a large number of successive works, especially on the phase transitions in various black hole solutions [5-40]. The recent progresses of these topics can be found in Ref. [41] for excellent reviews.

      One of the interesting black hole phase transitions is the famous Hawking–Page (HP) phase transition [42], which was originally studied between a Schwarzschild–AdS black hole and the thermal AdS vacuum. In the AdS space, a large black hole (with large horizon radius) has positive heat capacity and is thermodynamically stable, so it can be in equilibrium with the thermal background. The partition function of the black hole–thermal AdS system is dominated by the black hole phase at high temperatures, and by the thermal AdS phase in the low temperature limit. As a result, below a certain temperature (i.e., the HP temperature $ T_{\rm{HP}} $), no black hole solution exists anymore, and above $ T_{\rm{HP}} $, the thermal AdS gas will collapse into a stable large black hole. This phase transition was later widely studied in Refs. [43-56], also with emphasis on the extended phase space in Refs. [57-61].

      The aim of this paper is to investigate the HP phase transitions of the charged AdS black holes surrounded by quintessence dark energy (RN–qAdS black hole hereafter). In the last two decades, various cosmological observations have strongly revealed the accelerated expansion of our universe, and the underlying reason is usually named as dark energy. One of the most promising dynamical candidates of dark energy is the quintessence field, which is essentially a slowly rolling scalar field. The equation of state of quintessence reads $ p = w\rho $, with $ p $, $ \rho $, and $ w $ being the pressure, energy density, and state parameter respectively. In order to guarantee the cosmic acceleration and to satisfy the energy condition, the state parameter should lie in the range $ -1<w<-1/3 $. For the relevant issues of black holes surrounded by quintessence, see Refs. [62-80]. For example, in Ref. [80], the authors studied the effect of quintessence on the HP phase transitions for the Schwarzschild–AdS black holes and showed that quintessence always decreases the HP temperature.

      The basic motivations of our study of the HP phase transitions of the RN–qAdS black holes are twofold. First, there are critical phenomena only for the charged AdS black holes, but not for the neutral Schwarzschild–AdS black holes, because the critical pressure, volume, and temperature are all the functions of black hole charge [3]. Second, to take quintessence into account is not only for its effect on the cosmic acceleration, but also for its competition against the AdS background. Due to the negative cosmological constant, the AdS background offers a positive pressure and acts as a box in the universe. On the contrary, quintessence has a negative pressure and thus provides a repelling force effectively. Therefore, it is quite interesting to simultaneously compare these two opposite ingredients and to explore their overall influences on the HP phase transitions.

      This paper is organized as follows. In Sect. 2, we briefly list the thermodynamic properties of the RN–qAdS black holes in the extended phase space. In Sect. 3, the HP phase transitions are first analytically studied in a special case with $ w = -2/3 $. In Sect. 4, the critical pressure, electric potential, and normalization factor are carefully discussed in order. Based on these critical values, the HP phase transitions with general state parameter $ w $ are investigated in Sect. 5. We conclude in Sect. 6. In this paper, we work in the natural system of units and set $ c = G_{\rm N} = \hbar = k_{\rm B} = 1 $.

    Ⅱ.   HP PHASE TRANSITION IN THE EXTENDED PHASE SPACE
    • In this section, we first outline the thermodynamic properties of the RN–qAdS black hole in the extended phase space, and then discuss the HP phase transition in more detail.

    • A.   Thermodynamics of the RN–qAdS black hole

    • We start from the action of the RN–qAdS black hole [63],

      $ \frac{1}{16\pi}\int {{\rm{d}}}^4x\,\sqrt{-g} \left[ \left(R+ \frac{6}{l^2}-F_{\mu\nu}F^{\mu\nu} \right)+{\cal L}_{\rm q} \right], $

      where $ R $ is the Ricci scalar, $ F_{\mu\nu} $ is the electromagnetic tensor, and $ {\cal L}_{\rm q} $ is the Lagrangian of quintessence dark energy viewed as a barotropic perfect fluid, $ {\cal L}_{\rm q} = -\rho[1+w\ln({\rho}/{\rho_0})] $, with $ \rho_0 $ being an integral constant. Consequently, the metric of the RN–qAdS black hole reads [62]

      $ {\rm d} s^2 = f(r)\,{\rm d} t^2-\frac{{\rm d} r^2}{f(r)}-r^2\,{\rm d}\theta^2-r^2 \sin^2{\theta}\, {\rm d} \phi^2, $

      where

      $ f(r) = 1-\frac{2M}{r}+\frac{Q^2}{r^2}+\frac{8\pi p r^2}{3}-\frac{a}{r^{1+3w}}, $

      (2)

      with $ M $ and $ Q $ being the black hole mass and charge. The energy density of quintessence is normalized as

      $ \rho = - \frac{3wa}{2r^{3(1+w)}}, $

      (3)

      where $ a $ is a positive normalization factor. It will be shown later that there exists an upper bound for $ a $, if the HP phase transition can happen. From Eq. (2), the event horizon radius $ r_+ $ is determined as the largest root of $ f(r_{+}) = 0 $. Then, the RN–qAdS black hole mass can be expressed in terms of $ r_+ $,

      $ M = \frac{r_+}{2} \left(1+\frac{Q^2}{r_+^2}+\frac{8\pi p r_+^2}{3}-\frac{a}{r_+^{1+3w}} \right). $

      To avoid naked singularity, $ r_+ $ must be positive, and this sets the lower bound of $ M $ as $ M>Q $.

      Furthermore, the entropy of the RN–qAdS black hole can be obtained by the Bekenstein–Hawking formula as one quarter of the event horizon area $ A $ [1,2],

      $ S = \frac{A}{4} = \pi r_+^2. $

      (4)

      From Eqs. (1) and (4), the RN–qAdS black hole mass can be reexpressed as a function of the thermodynamic variables $ S $, $ p $, $ Q $, and the normalization factor $ a $,

      $ M = \frac 12\sqrt{ \frac S\pi} \left[1+ \frac{\pi Q^2}{S}+ \frac{8pS}{3}-a \left( \frac \pi S \right)^{ \frac{1+3w}{2}} \right]. $

      (5)

      The first law of thermodynamics for the RN–qAdS black hole in the extended phase space can be obtained by a direct differentiation of Eq. (5),

      $ {{\rm{d}}} M = T\, {{\rm{d}}} S+V\, {{\rm{d}}} p+\Phi\, {{\rm{d}}} Q, $

      (6)

      where $ T $, $ V $, and $ \Phi $ are the Hawking temperature, thermodynamic volume, and electric potential at the event horizon of the RN–qAdS black hole respectively,

      $ T = \left( \frac{ \partial M}{ \partial S} \right)_{p,Q} = \frac{1}{4\sqrt{\pi S}} \left[1- \frac{\pi Q^2}{S}+8pS+3wa \left(\frac{\pi}{S} \right)^{ \frac{1+3w}{2}} \right], $

      (7)

      $ V = \left( \frac{ \partial M}{ \partial p} \right)_{S,Q} = \frac{4\pi}{3} \left( \frac S\pi \right)^{ \frac 32}, $

      (8)

      $ \Phi = \left( \frac{ \partial M}{ \partial Q} \right)_{S,p} = Q\sqrt{ \frac\pi S}. $

      (9)

      Moreover, these results can be consistently rewritten in terms of $ r_+ $ as

      $ T = \frac{f'(r_+)}{4\pi}, \quad V = \frac{4\pi r_+^3}{3}, \quad \Phi = \frac{Q}{r_+}. $

      By this means, Eq. (7) becomes

      $ T = \frac{1}{4\sqrt{\pi S}} \left[1-\Phi^2+8pS+3wa \left(\frac{\pi}{S} \right)^{ \frac{1+3w}{2}} \right]. $

      (10)

      In addition, we find from Eq. (6) that the $ p $$ V $ term has the form $ V\, {{\rm{d}}} p $, but not $ -p\, {{\rm{d}}} V $. Therefore, the RN–qAdS black hole mass $ M $ should be identified as its enthalpy rather than internal energy in the extended phase space.

      Last, the Smarr relation (i.e., the Gibbs–Duhem relation in traditional thermodynamics) can be obtained by a scaling argument as $ M = 2TS-2pV+\Phi Q $, consistent with the first law of black hole thermodynamics. In fact, this is one of the basic motivations to study the extended phase space, in which the cosmological constant is varying, and the scaling argument thus applies.

      In our present work, the normalization factor $ a $ is understood as a fixed parameter characterizing the energy density of quintessence field. While, there were also alternative interpretations of $ a $ as a thermodynamic variable [18,81-83]. In this way, the first law of black hole thermodynamics and the Smarr relation are generalized to $ {{\rm{d}}} M = T\, {{\rm{d}}} S+V\, {{\rm{d}}} p+\Phi\, {{\rm{d}}} Q+{\cal A}\, {{\rm{d}}} a $ and $ M = 2TS-2pV+ \Phi Q+(1+3w){\cal A}a $, with $ {\cal A} $ being the conjugate variable of $ a $, $ {\cal A} = \left({ \partial M}/{ \partial a} \right)_{p,Q,S} = - \left({\pi}/{S} \right)^{{3w}/{2}}/2 $. However, we should stress that the specific forms of the first law and the Smarr relation depend on the choice of thermodynamic variables, and whether or not to choose $ a $ as a thermodynamic variable is not unique. Nevertheless, we will see that the definition of the Gibbs free energy are the same in our work and Refs. [18,81-83], so the relevant discussions of the HP phase transitions are consistent in between, not altered by the interpretation of the normalization factor $ a $.

    • B.   HP phase transition

    • Before the discussion of the HP phase transition, one important issue should be clarified in advance. Due to the conservation of charge, a black hole with fixed charge cannot undergo the HP phase transition to the thermal AdS vacuum that is electrically neutral. Consequently, the HP phase transitions of the RN–qAdS black holes will be studied in the grand canonical ensemble, in which the electric potential $ \Phi $ is fixed, and the electric charge $ Q $ is thus allowed to vary.

      Since in the extended phase space the black hole mass is regarded as enthalpy, the corresponding thermodynamic potential in the grand canonical ensemble should be the Gibbs free energy,

      $ G(T,p,\Phi) = M-TS-\Phi Q. $

      (11)

      Substituting Eqs. (5), (9), and (10) into Eq. (11), we obtain

      $ G = \frac{1}{4}\sqrt{\frac{S}{\pi}} \left[1-\Phi^2-\frac{8pS}{3}-(2+3w)a \left(\frac{\pi}{S} \right)^{ \frac{1+3w}{2}} \right]. $

      (12)

      Meanwhile, since the total number of the thermal gas particles in the AdS space is not conserved and can vary with temperature, the Gibbs free energy of the thermal AdS background is always 0.

      Therefore, the criterion of the HP phase transition is that the Gibbs free energy of the black hole–thermal AdS system vanishes,

      $ G = 0, $

      (13)

      and the HP temperature $ T_{\rm{HP}} $ is then fixed by Eq. (13) accordingly. In the following sections, the Gibbs free energy of the RN–qAdS black hole will be shown to monotonically decrease with temperature and will be negative above $ T_{\rm{HP}} $. Therefore, below $ T_{\rm{HP}} $, the thermal AdS phase with vanishing $ G $ is more stable; above $ T_{\rm{HP}} $, the black hole phase with negative $ G $ is more preferred, and the thermal AdS gas will collapse into the black hole. This interesting behavior indicates that the thermal AdS vacuum is more like a solid rather than an ordinary gas in the HP phase transition [84].

    Ⅲ.   HP PHASE TRANSITIONS WITH w=−2/3
    • The basic problems to be studied in the HP phase transitions include two aspects: to determine the HP temperature $ T_{\rm{HP}} $ and to figure out the Gibbs free energy $ G $ of the black hole–thermal AdS system as a function of temperature. First, at the HP phase transition point, we should substitute Eq. (12) into the criterion in Eq. (13) to obtain the black hole entropy $ S $ in terms of $ p $, $ \Phi $, and $ a $, and then substitute $ S(p,\Phi,a) $ into Eq. (10) to obtain $ T_{\rm{HP}} $. Next, we solve $ S $ from Eq. (10) as a function of $ T $, and then substitute it into Eq. (12) to obtain the $ G $$ T $ relation of the RN–qAdS black hole at arbitrary temperature, so as to achieve the global phase structure of the HP phase transition. Altogether, we choose the black hole entropy $ S $ as the intermediate variable in all the following calculations. Of course, it is equivalent to choosing the event horizon radius $ r_+ $ instead of $ S $, as their simple relation is shown in Eq. (4).

      Generally speaking, the calculation of the RN–qAdS black hole entropy is difficult, as from the complicated expression in Eq. (12), the analytical result is usually not available, unless the state parameter $ w $ takes some special value. However, as we see from the last term in Eq. (12), when $ w = -2/3 $ [i.e., the middle point in its range $ (-1,-1/3) $], the influence from quintessence on the Gibbs free energy vanishes, and the corresponding calculations will be greatly simplified. Therefore, we first explicitly analyze the special case with $ w = -2/3 $ for mathematical clarity in this section, investigating the effects of quintessence on the HP phase transitions, and move on to the more general cases in the following sections.

      When $ w = -2/3 $, the relevant thermodynamic quantities in Eqs. (10) and (12) reduce to

      $ T = \frac{1}{4\pi} \left[\sqrt{\frac{\pi}{S}}(1-\Phi^2+8pS)-2a \right], $

      (14)

      $ G = \frac{1}{4}\sqrt{\frac{S}{\pi}} \left(1-\Phi^2-\frac{8pS}{3} \right). $

      (15)

      We observe from these results that, although quintessence does not affect the Gibbs free energy in the present circumstance, it does decrease the black hole temperature.

    • A.   HP temperature

    • At the HP phase transition point, $ G $ vanishes in Eq. (15), so we have $ S = 3(1-\Phi^2)/(8p) $. Substituting $ S $ into Eq. (14), we obtain the HP temperature $ T_{\rm{HP}} $,

      $ T_{\rm{HP}} = \sqrt{\frac{8p}{3\pi}(1-\Phi^2)}-\frac{a}{2\pi}. $

      (16)

      From Eq. (16), we clearly see that quintessence decreases $ T_{\rm{HP}} $, despite the fact that $ G $ is now irrelevant to $ a $. If the HP phase transition happens, $ T_{\rm{HP}} $ should be positive, and this requirement sets the corresponding bounds for $ p $, $ \Phi $, and $ a $ respectively,

      $ p>p_{\min} = \dfrac{3a^2}{32\pi(1-\Phi^2)}, $

      (17)

      $ \Phi<\Phi_{\max} = \sqrt{1- \displaystyle\frac{3a^2}{32\pi p}}, $

      (18)

      $ a<a_{\max} = \sqrt{\displaystyle\frac{32\pi p}{3}(1-\Phi^2)}. $

      (19)

      The $ T_{\rm{HP}}-p $ curves of the RN–qAdS black holes with different values of $ \Phi $ and $ a $ are shown in Figs. 1 and 2. We find that the HP phase transitions can happen at all high pressures, as there is no terminal point in the $ T_{\rm{HP}} $$ p $ curves (i.e., the coexistence lines) in the phase diagram. In the low pressure limit, $ p $ has the lower bound $ p_\min $. More interestingly, the thermal AdS phase lies below the coexistence line, meaning that it behaves like a solid in the HP phase transition. The HP temperature $ T_{\rm{HP}} $ is found to increase with $ p $ and to decrease with $ \Phi $ and $ a $, consistent with the previous analyses in Ref. [61].

      Figure 1.  The HP temperature $T_{\rm{HP}}$ as a function of pressure $p$, with different electric potentials $\Phi$ and fixed normalization factor $a = 1$. $T_{\rm{HP}}$ increases with $p$ and decreases with $\Phi$.

      Figure 2.  The HP temperature $T_{\rm{HP}}$ as a function of pressure $p$, with different normalization factors $a$ and fixed electric potential $\Phi = 0.5$. $T_{\rm{HP}}$ increases with $p$ and decreases with $a$.

    • B.   Gibbs free energy

    • Next, at arbitrary temperature, we first solve entropy $ S $ from Eq. (14),

      $\begin{aligned} S = & \frac{1}{32\pi p^{2} }\Big[(2 \pi T+a)^{2}-4\pi p(1-\Phi^{2}) \\ & \pm (2 \pi T+a)\sqrt{(2 \pi T+a)^2-8\pi p(1-\Phi^{2})}\Big]. \end{aligned} $

      (20)

      There are two branches of $ S $ in Eq. (20), corresponding to the stable large and unstable small black hole solutions respectively. Substituting Eq. (20) into Eq. (15), we obtain the Gibbs free energies of the large and small RN–qAdS black holes as a function of $ T $, $ p $, $ \Phi $, and $ a $,

      $ \begin{aligned} G = & -\frac{1}{192\sqrt{2}\pi^2p^2}\Big[(2 \pi T+a)^2-4\pi p(1-\Phi^2) \\ & \pm(2 \pi T+a) \sqrt{(2 \pi T+a)^{2}-8\pi p(1-\Phi^{2})}\Big]^{ \frac 12} \\ & \times \Big[(2 \pi T+a)^{2}-16 \pi p(1-\Phi^2) \\ & \pm(2 \pi T+a) \sqrt{(2 \pi T+a)^{2}-8 \pi p(1-\Phi^{2})}\Big]. \end{aligned} $

      The $ G $$ T $ curves of the stable large and unstable small RN–qAdS black holes with different values of $ p $, $ \Phi $, and $ a $ are shown in Figs. 35. The $ G $$ T $ curves of the stable large RN–qAdS black holes intersect the $ T $-axis at the HP temperature $ T_{\rm{HP}} $. In Fig. 3, the intersection points move rightward, indicating that $ T_{\rm{HP}} $ increases with $ p $. Similarly, in Figs. 4 and 5, $ T_{\rm{HP}} $ decreases with $ \Phi $ and $ a $, as already shown in Figs. 1 and 2. Below or above $ T_{\rm{HP}} $, the thermal AdS phase or the stable large black hole phase is globally preferred respectively. The $ G $$ T $ curves of the unstable small RN–qAdS black holes are always above the $ T $-axis, so the HP phase transition never happens for them.

      Figure 3.  The Gibbs free energies $G$ of the large and small RN–qAdS black holes as a function of temperature $T$, with different pressures $p$ and fixed electric potential $\Phi = 0.5$ and normalization factor $a = 1$. The HP temperature $T_{\rm{HP}}$ increases with $p$.

      Figure 4.  The Gibbs free energies $G$ of the large and small RN–qAdS black holes as a function of temperature $T$, with different electric potentials $\Phi$ and fixed pressure $p = 1$ and normalization factor $a = 1$. The HP temperature $T_{\rm{HP}}$ decreases with $\Phi$.

      Figure 5.  The Gibbs free energies $G$ of the large and small RN–qAdS black holes as a function of temperature $T$, with different normalization factors $a$ and fixed pressure $p = 1$ and electric potential $\Phi = 0.5$. The HP temperature $T_{\rm{HP}}$ decreases with $a$.

    Ⅳ.   CRITICAL VALUES WITH GENERAL $ w $
    • We continue to discuss the HP phase transitions of the RN–qAdS black holes with general state parameter $ w $ of quintessence. This generalization is physically straightforward but mathematically tedious. In most circumstances, the analytical results are unavailable due to the exponents $ (1+3w)/2 $ in Eqs. (10) and (12). Therefore, before going into details, we first consider the critical situation, when $ T_{\rm{HP}} = 0 $ and $ G = 0 $ simultaneously in Eqs. (10) and (12). Under such conditions, by eliminating the variable $ S $, we obtain the following constraint of $ p $, $ \Phi $, $ a $, and $ w $,

      $ \frac{3a(1+w)}{2(1-\Phi^2)} = \left[-\frac{(1+3w)(1-\Phi^2)}{8\pi p(1+w)} \right]^{ \frac{1+3w}{2}}. $

      From the above constraint, the expressions of the critical values $ p_{\rm c} $, $ \Phi_{\rm c} $, and $ a_{\rm c} $ with general state parameter $ w $ can be easily obtained as

      $ p_{\rm c} = -\frac{1+3w}{8\pi} \left(\frac{2}{3a} \right)^{ \frac{2}{1+3w}} \left(\frac{1-\Phi^2}{1+w} \right)^{ \frac{3(1+w)}{1+3w}}, $

      (21)

      $ \Phi_{\rm c} = \sqrt{1-(1+w) \left(\frac{3a}{2} \right)^{ \frac{2}{3(1+w)}} \left(-\frac{8\pi p}{1+3w} \right)^{ \frac{1+3 w}{3(1+ w)}}}, $

      (22)

      $ a_{\rm c} = \frac{2}{3} \left(-\frac{1+3w}{8 \pi p} \right)^{ \frac{1+3w}{2}} \left(\frac{1-\Phi^2}{1+w} \right)^{ \frac{3(1+w)}{2}}. $

      (23)

      These critical values are the natural extension of the results in Eqs. (17)–(19). Below, we discuss their dependence on $ w $ in order, with special emphases on two limiting cases, $ w\to-1 $ and $ w\to-1/3 $, where simple and analytical expressions exist. For general $ w $, the results are shown in Figs. 68 by numerical methods.

      Figure 6.  The critical pressure $p_{\rm c}$ as a function of the state parameter $w$, with different normalization factors $a$ and fixed electric potential $\Phi = 0.5$. When $w\to-1$, the limit of $p_{\rm c}$ is finite for any $\Phi$ and $a$. When $w\to -1/3$, if $a<0.75$, $p_{\rm c}$ monotonically decreases, with the limit $p_{\rm c}\to 0$; if $a>0.75$, $p_{\rm c}$ first decreases to the minimum and then increases to infinity. The curve with $a = 0.75$ is the boundary between these two branches. The HP phase transition can happen only in the region $p>p_{\rm c}$.

      Figure 7.  The critical electric potential $\Phi_{\rm c}$ as a function of the state parameter $w$, with different normalization factors $a$ and fixed pressure $p = 1$. When $w\to-1$, $\Phi_{\rm c}$ first increases and then decreases with $w$ if $a>8\pi/3$, or monotonically decreases with $w$ if $a<8\pi/3$. When $w\to-1/3$, $\Phi_{\rm c}$ monotonically decreases, but $w$ cannot reach $-1/3$ unless $a<1$. The HP phase transition can happen only in the region $\Phi<\Phi_{\rm c}$.

      Figure 8.  The critical normalization factor $a_{\rm c}$ as a function of the state parameter $w$, with different pressures $p$ and fixed electric potential $\Phi = 0.5$. There always exists an upper bound for $a_{\rm c}$ when $-1<w<-1/3$, and the HP phase transition can happen only in the region $a<a_{\rm c}$.

    • A.   Critical pressure

    • For the critical pressure $ p_{\rm c} $, in the limit $ w\to -1 $, Eq. (21) reduces to

      $ p_{\rm c}\to \frac{3a}{8\pi} \left( \frac{1+w}{1-\Phi^2} \right)^{ \frac{3(1+w)}{2}}. $

      In this case, $ p_{\rm c} $ is bounded and will reach a finite value $ 3a/(8\pi) $, independent of $ \Phi $.

      In the limit $ w\to-1/3 $, Eq. (21) reduces to

      $ p_{\rm c}\to-\frac{1+3w}{8\pi} \left( \frac{1-\Phi^2}{a} \right)^{ \frac{2}{1+3w}}. $

      In this case, the situation becomes more complex, and the complexity origins from the exponent $ 2/(1+3w) $. When $ w\to -1/3 $, this exponent goes to negative infinity. Therefore, the limit of $ p_{\rm c} $ depends on the ratio $ (1-\Phi^2)/a $. If $ a< 1-\Phi^2 $, the ratio is greater than 1, so $ p_{\rm c}\to 0 $. In contrast, if $ a>1-\Phi^2 $, the ratio is less than 1, so $ p_{\rm c}\to\infty $.

      The $ p_{\rm c} $$ w $ curves are shown in Fig. 6. The HP phase transition can happen only when the pressure $ p $ is above the critical value $ p_{\rm c} $ for a given $ w $, so there is always a minimum pressure in any $ T_{\rm{HP}} $$ p $ curve in Figs. 1 and 2.

    • B.   Critical electric potential

    • Next, for the critical electric potential $ \Phi_{\rm c} $, in the limit $ w\to -1 $, Eq. (22) reduces to

      $ \Phi_{\rm c}\to\sqrt{1-(1+w) \left(\frac{3a}{8\pi p} \right)^{ \frac{2}{3(1+w)}}}. $

      Now, the exponent $ 2/[3(1+w)] $ will reach positive infinity. Therefore, a similar complexity as that in Sect. 4.1 appears. If $ a<8\pi p/3 $, $ \Phi_{\rm c} $ decreases with $ w $, with the limit $ \Phi_{\rm c}\to 1 $. However, if $ a>8\pi p/3 $, the situation is more complicated. Because the expression in the square root must be positive, the minimum of $ w $ actually cannot reach $ -1 $. In this case, $ \Phi_{\rm c} $ first rapidly increases and then decreases with $ w $.

      In the limit $ w\to -1/3 $, Eq. (22) reduces to

      $ \Phi_{\rm c}\to\sqrt{1-a \left(-\frac{8\pi p}{1+3w} \right)^{ \frac{1+3w}{2}}}. $

      Again, the expression in the square root should be positive. It is not difficult to see $ \lim\limits_{w\to-1/3}\big(-\displaystyle\frac{8\pi p}{1+3w}\big)^{ \displaystyle\frac{1+3w}{2}} = 1 $, and this limit is independent of $ p $. Therefore, if $ a<1 $, $ w $ can reach $ -1/3 $, but if $ a>1 $, $ w $ actually cannot reach $ -1/3 $.

      The $ \Phi_{\rm c} $$ w $ curves are shown in Fig. 7. The HP phase transition can happen only when the electric potential $ \Phi $ is below the critical value $ \Phi_{\rm c} $ for a given $ w $.

    • C.   Critical normalization factor

    • Last, for the critical normalization factor $ a_{\rm c} $, in the limit $ w\to-1 $, Eq. (23) reduces to

      $ a_{\rm c}\to\frac{8\pi p}{3} \left(\frac{1-\Phi^2}{1+w} \right)^{ \frac{3(1+w)}{2}}. $

      In this case, $ a_{\rm c} $ is bounded and will reach a finite value $ 8\pi p/3 $, independent of $ \Phi $.

      In the limit $ w\to-1/3 $, Eq. (23) reduces to

      $ a_{\rm c}\to(1-\Phi^2) \left(-\frac{1+3w}{8 \pi p} \right)^{ \frac{1+3w}{2}}. $

      Similarly, in this case, $ a_{\rm c} $ is also bounded and will reach a finite value $ 1-\Phi^2 $, independent of $ p $.

      The relation of $ a_{\rm c} $ and $ w $ needs more careful inspection. From the above two limiting cases, we find that $ a_{\rm c} $ first increases with $ w $, reaching the maximum point at $ (w_0,a_{\rm c0}) $, and then decreases with $ w $. The exact values of $ w_0 $ and $ a_{\rm c0} $ can be obtained by setting $ {\partial a_{\rm c}}/{\partial w} = 0 $ in Eq. (23),

      $ w_0 = -\frac{1-\Phi^2+8\pi p}{3(1-\Phi^2)+8\pi p},\quad a_{\rm c0} = 1-\Phi^2+\frac{8\pi p}{3}. $

      (24)

      These results indicate that with fixed $ \Phi $ and $ p $, if the HP phase transition happens, $ a_{\rm c} $ should have an upper bound $ a_{\rm c0} $. Then, it is easy to see $ -1<w_0<-1/3 $, meaning that this upper bound always exists for quintessence. Altogether, $ a_{\rm c} $ is not monotonic when $ -1<w<-1/3 $, and this result is very important, when we analyze the effects of quintessence on the HP temperature $ T_{\rm{HP}} $ in the next section.

      The $ a_{\rm c} $$ w $ curves are shown in Fig. 8. The HP phase transition can happen only when the normalization factor $ a $ is below the critical value $ a_{\rm c} $ for a given $ w $.

      Till now, we are allowed to summarize our basic conclusion of the critical values. For a given state parameter $ w $, the HP phase transition can happen only with large pressure $ p $ and small electric potential $ \Phi $ and normalization factor $ a $ (i.e., $ p>p_{\rm c} $, $ \Phi<\Phi_{\rm c} $, and $ a<a_{\rm c} $). Otherwise, the Gibbs free energy of the stable large RN–qAdS black hole is always less than that of the thermal AdS vacuum, and there will be no possibility of the HP phase transition accordingly.

    Ⅴ.   HP PHASE TRANSITIONS WITH GENERAL $ w $
    • Finally, we move on to study the HP phase transitions of the RN–qAdS black holes in the extended phase space, with the most general state parameter $ -1<w<-1/3 $. In principle, the relevant calculations are in parallel to those in Sect. 3 with the special value $ w = -2/3 $. However, for general $ w $, the last term in Eq. (12) can no longer be disregarded, making it impossible to obtain the analytical results any more. Therefore, we perform the calculations and illustrate our results totally by numerical methods in this section.

    • A.   HP temperature

    • To see the effects of quintessence on the HP temperature, we first show the $ T_{\rm{HP}} $$ p $ curves in Fig. 9, with different $ a $ and fixed $ \Phi $ and $ w $. The curves are similar to those in Fig. 2 with $ w = -2/3 $. The HP temperature $ T_{\rm{HP}} $ increases with pressure $ p $, and $ p $ should be greater than its critical value $ p_{\rm c} $. At a fixed pressure, $ T_{\rm{HP}} $ decreases with $ a $, meaning that quintessence diminishes the HP temperature and thus induces the HP phase transition from the thermal AdS phase to the black hole phase at low temperatures. Therefore, $ a $ always has a maximum $ a_{\rm c} $, since $ T_{\rm{HP}} $ should be positive.

      Figure 9.  The HP temperature $T_{\rm{HP}}$ as a function of pressure $p$, with different normalization factors $a$ and fixed electric potential $\Phi = 0.5$ and state parameter $w = -11/30$. $T_{\rm{HP}}$ decreases with $a$, indicating that quintessence always tends to induce the HP phase transition at low temperatures.

      Next, we show the $ T_{\rm{HP}} $$ p $ curves in Fig. 10, with different $ w $ and fixed $ \Phi $ and $ a $. Again, $ T_{\rm{HP}} $ increases with pressure $ p $, and $ p>p_{\rm c} $. In most cases, at a fixed pressure, $ T_{\rm{HP}} $ decreases with $ w $. However, it must be pointed out that, when $ w $ is close to $ -1 $, two $ T_{\rm{HP}} $$ p $ curves can even intersect, and this intersection will lead to some unusual consequences. First, two quintessence fields with different $ w $ but fixed $ p $, $ \Phi $, and $ a $ can have the same HP temperature $ T_{\rm{HP}} $. Second, when $ w $ is near $ -1 $, $ T_{\rm{HP}} $ turns out to increase with $ w $ (all to be explained in more detail in Fig. 11).

      Figure 10.  The HP temperature $T_{\rm{HP}}$ as a function of pressure $p$, with different state parameters $w$ and fixed electric potential $\Phi = 0.5$ and normalization factor $a = 1$. In most cases, $T_{\rm{HP}}$ decreases with $w$, but when $w$ is close to $-1$, the curves with different $w$ can intersect, and $T_{\rm{HP}}$ may increase with $w$.

      Figure 11.  The HP temperature $T_{\rm{HP}}$ as a function of normalization factor $a$, with different state parameters $w$ and fixed pressure $p = 1$ and electric potential $\Phi = 0.5$. All curves start from the same point at $(0,0.798)$. When $w = -2/3$, the $T_{\rm{HP}}$$a$ curve is straight, as indicated in Eq. (16). The curve with $w = w_0 = -0.945$ has the largest intercept $a_{\rm c0} = 9.128$ on the $a$-axis. The different curves (e.g., the one with $w = -8/9$ and the one with $w = -0.999$) can intersect each other. Moreover, it can be seen that $T_{\rm{HP}}$ usually decreases with $w$, but may also increase with $w$ when $w$ is very close to -1.

      Below, we further investigate the dependence of $ T_{\rm{HP}} $ on the two parameters of quintessence, $ a $ and $ w $. For simplicity, we show the $ T_{\rm{HP}} $$ a $ curves in Fig. 11, with different $ w $ and fixed $ p $ and $ \Phi $. When $ a = 0 $ (i.e., without quintessence), $ T_{\rm{HP}} $ should be entirely determined by $ p $ and $ \Phi $. Therefore, all the $ T_{\rm{HP}} $$ a $ curves set out from the same starting point at $ (0,\sqrt{8p(1-\Phi^2)/(3\pi)}) $. When $ a\neq 0 $, the situation is subtle and needs more explanation. As explained in Eq. (24), when $ w = w_0 $, the corresponding $ T_{\rm{HP}} $$ a $ curve has the largest intercept $ a_{\rm c0} $ on the $ a $-axis. Hence, one $ T_{\rm{HP}} $$ a $ curve with $ w<w_0 $ can intersect another one with $ w>w_0 $ in Fig. 11, meaning that two RN–qAdS black holes with the same $ p $, $ \Phi $, and $ a $ but different $ w $ can have the same HP temperature $ T_{\rm{HP}} $. Furthermore, when $ w $ is even smaller and $ a $ is large enough, $ T_{\rm{HP}} $ begins to increase with $ w $, as already seen in Fig. 10.

    • B.   Gibbs free energy

    • Here, we show the $ G $$ T $ curves of the stable large and unstable small RN–qAdS black holes in Fig. 12. As the cases with different values of $ p $, $ \Phi $, and $ a $ have already been carefully studied in Sect. 3.2 when $ w = -2/3 $, we need not repeat them here, as the relevant results are qualitatively analogous for arbitrary $ w $. Therefore, we only concentrate our attention to the Gibbs free energy with the variation of the state parameter $ w $, and fix other parameters $ p $, $ \Phi $, and $ a $. With the special values of $ p $, $ \Phi $, and $ a $ in Fig. 12, the HP temperature $ T_{\rm{HP}} $ monotonically decreases with $ w $, consistent with the previous analysis in Fig 11.

      Figure 12.  The Gibbs free energies $G$ of the large and small RN–qAdS black holes as a function of temperature $T$, with different state parameters $w$ and fixed pressure $p = 1$, electric potential $\Phi = 0.5$, and normalization factor $a = 1$. The HP temperature $T_{\rm{HP}}$ decreases with $w$.

    • C.   Discussion

    • Last, we discuss the effects of quintessence on the HP temperature $ T_{\rm{HP}} $ in general. With a negative pressure, quintessence field acts as a cold environment around the black hole. This property is exactly the opposite to that of the AdS vacuum, which offers a positive pressure. Since $ T_{\rm{HP}} $ generally increases with pressure, the more quintessence influences, the lower $ T_{\rm{HP}} $ will be.

      Let us look at the normalization condition in Eq. (3) at the event horizon radius $ r_+ $. The effects of quintessence are encoded in two parameters, $ a $ and $ w $,

      $ \rho = - \frac{3wa}{2r_+^{3(1+w)}}. $

      (25)

      It is easy to see that the energy density $ \rho $ of quintessence is proportional to the normalization factor $ a $, so $ T_{\rm{HP}} $ always decreases with $ a $, as already shown in Figs. 9 and 11.

      However, the dependence of $ T_{\rm{HP}} $ on $ w $ is more intricate. Generally speaking, when $ a $ is small, $ T_{\rm{HP}} $ monotonically decreases with $ w $, but when $ a $ is large enough, $ T_{\rm{HP}} $ first increases and then decreases with $ w $. We summarize these results in a 3-dimensional plot of $ T_{\rm{HP}} $ as a function of $ w $ and $ a $ in Fig. 13, with fixed pressure and electric potential. The projection of the counters with different $ a $ on the $ T_{\rm{HP}} $$ w $ plane explicitly exhibits the monotonicity of the $ T_{\rm{HP}} $$ w $ curves.

      Figure 13.  The HP temperature $T_{\rm{HP}}$ as a function of the state parameter $w$ and normalization factor $a$, with fixed pressure $p = 1$ and electric potential $\Phi = 0.5$. The $T_{\rm{HP}}$$w$ curves can be directly produced by a projection of the counters with 8 different values of $a = 0$, 0.5, 1, 2, 4, 7, $8\pi/3$, and 8.8, and the monotonicity of the $T_{\rm{HP}}$$w$ curves can be observed explicitly. When $a<1$, $T_{\rm{HP}}$ monotonically decreases with $w$. When $1<a<8\pi/3$, $T_{\rm{HP}}$ also monotonically decreases with $w$, but $w$ can no longer reach $-1/3$. When $a>8\pi/3$, $T_{\rm{HP}}$ first increases and then decreases with $w$.

      It seems counterintuitive that $ T_{\rm{HP}} $ may even decrease with $ w $, because as $ w $ increases, the effect of quintessence becomes weaker, and $ T_{\rm{HP}} $ seems to increase accordingly, but actually it drops. However, there is no ambiguity at all. In fact, the dependence of $ \rho $ on $ w $ in Eq. (25) is not very simple, since there are $ w $ both in the numerator and denominator, so the relation of $ \rho $ and $ w $ also depends on $ r_+ $. By Eq. (4), we can rewrite Eq. (25) as

      $ \rho = - \frac{3wa}{2} \left( \frac\pi S \right)^{ \frac{3(1+w)}{2}}. $

      Hence, when discussing the relation of $ \rho $ and $ w $, we must take the special value of $ S $ into account.

      At the HP phase transition point, the corresponding entropy $ S $ should be determined by setting $ G = 0 $ in Eq. (12),

      $ 1-\Phi^2-\frac{8pS}{3}-(2+3w)a \left(\frac{\pi}{S} \right)^{ \frac{1+3w}{2}} = 0. $

      This equation is not analytically solvable, and we focus on the limiting cases with $ w\to-1/3 $ and $ w\to-1 $ respectively. First, when $ w\to-1/3 $, we have

      $ S\to \frac{3(1-\Phi^2-a)}{8p}. $

      If $ p = 1 $, $ \Phi = 0.5 $, and $ a $ is small, we find that $ S $ is also a small quantity around 0.2. In this case, $ \rho $ is an increasing function of $ w $, so the influence from quintessence actually becomes stronger when $ w\to-1/3 $, and the HP temperature $ T_{\rm{HP}} $ thus decreases with $ w $, as shown in Fig. 13. Second, when $ w\to-1 $, we have

      $ S\to \frac{3\pi(1-\Phi^2)}{8\pi p-3a}. $

      If $ p = 1 $, $ \Phi = 0.5 $, and $ a $ is large enough (e.g., $ a = 8 $ around $ a_{\rm c0} $), we find that $ S $ is now a large quantity around 6. Thus, $ \rho $ is a decreasing function of $ w $ this time, so $ T_{\rm{HP}} $ first increases with $ w $, and then decreases as before, when $ w $ is away from $ -1 $. This behavior can be directly seen in Fig. 13, also from the intersection of the $ T_{\rm{HP}} $$ p $ curves in Fig. 10 and the intersection of the $ T_{\rm{HP}} $$ a $ curves in Fig. 11.

    Ⅵ.   CONCLUSION
    • The HP transition between the AdS black hole and the thermal AdS vacuum is an important issue in black hole thermodynamics. In this paper, the HP phase transitions of the charged AdS black holes surrounded by quintessence are systematically investigated in the extended phase space. On the one hand, the negative varying cosmological constant in the AdS space presents a positive thermodynamic pressure. On the other hand, the quintessence field, with the state parameter $ -1<w<-1/3 $, offers a negative pressure. Consequently, the HP phase transition is an overall result of these two competitive factors.

      The quintessence field modifies the metric of the RN–AdS black hole, and thus influences black hole thermodynamics via introducing two new parameters, the normalization factor $ a $ and the state parameter $ w $. We calculate the HP temperature $ T_{\rm{HP}} $ as a function of $ p $, $ \Phi $, $ a $, and $ w $, and also the Gibbs free energy $ G $ as a function of $ T $, $ \Phi $, $ a $, and $ w $. All the calculations are performed first for the special case with $ w = -2/3 $ and then for general $ w $. The basic conclusions of our work can be drawn as follows:

      1. The HP temperature $ T_{\rm{HP}} $ increases with pressure $ p $, consistent with our previous work on the RN–AdS black holes in Ref. [61]. However, due to the existence of quintessence, $ p $ should be greater than its critical value $ p_{\rm c} $, such that $ T_{\rm{HP}} $ is positive.

      2. The Gibbs free energy $ G $ of the RN–qAdS black hole decreases with $ T $, so below or above $ T_{\rm{HP}} $, the thermal AdS phase or stable large RN–qAdS black hole phase is globally favored respectively, also consistent with Ref. [61].

      3. When the HP phase transition happens, for the sake of positive $ T_{\rm{HP}} $, the variables of $ p $, $ \Phi $, and $ a $ cannot exceed their critical values (i.e., $ p>p_{\rm c} $, $ \Phi<\Phi_{\rm c} $, and $ a<a_{\rm c} $). We plot $ p_{\rm c} $, $ \Phi_{\rm c} $, and $ a_{\rm c} $ as a function of $ w $ in order in Sect. 4.

      4. The dependence of $ T_{\rm{HP}} $ on the two parameters of quintessence, $ a $ and $ w $, is not quite so simple. First, $ T_{\rm{HP}} $ monotonically decreases with $ a $. However, $ T_{\rm{HP}} $ decreases with $ w $, only when $ a $ is small. If $ a $ is large enough, $ T_{\rm{HP}} $ first increases and then decreases with $ w $. All these complexities arise from the relation of $ \rho $ and $ w $ in the normalization condition in Eq. (3). Nevertheless, whenever the energy density $ \rho $ of quintessence increases, $ T_{\rm{HP}} $ drops, so we are allowed to claim that the quintessence field always decreases $ T_{\rm{HP}} $ and facilitates the HP phase transition at low temperatures. This conclusion is consistent with the result in Ref. [80] and includes Ref. [80] as a limiting case of our work with $ \Phi = 0 $.

      Altogether, we wish to present a whole picture of the HP phase transitions of the RN–qAdS black holes in the extended phase space. Of course, it is straightforward to generalize this study to the rotating and charged rotating AdS black holes with quintessence. However, from our experience in Ref. [61], except the unnecessary mathematical inconvenience caused by rotation, we do not expect much new physical insight to be achieved.

      We are very grateful to Bing-Yu Su, Yuan-Yuan Wang, and Zhi-Zhen Wang for fruitful discussions.

Reference (84)

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