# λ Phase Transition in Horava Gravity

Authors: Wei Xu
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We present another example of superfluid black hole containingλ phase transition in Horava gravity. After studying the extended thermodynamics of general dimensional Horava-Lifshitz AdS black holes, it is found that only the one with spherical horizon in four and five dimensions has λ phase transition, which is a line of (continuous) second-order phase transitions and was famous in the discussion of superfluidity of liquid He4. The “superfluid” black hole phase and “normal” black hole phase are also distinguished. Particularly, six-dimensional Horava-Lifshitz AdS black holes exhibit infinitely many critical points in P-ν plane and the divergent points for specific heat, for which they only contain the “normal” black hole phase and the “superfluid” black hole phase disappears due to the physical temperature constraint; therefore there is no similar phase transition. In more than six dimensions, there is no P-ν critical behavior. After choosing the appropriate ordering field, we study the critical phenomena in different planes of thermodynamical phase space. We also calculate the critical exponents, which are the same as the van der Waals fluid.


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## Body

## 1. Introduction

Black hole thermodynamics always provides valuable insight into quantum properties of gravity, and it has been studied extensively for quite a long time, especially for the quantum and microscopic interpretation of black hole temperature and entropy (see [1, 2], for example). Besides, thermodynamics and phase transitions of AdS black holes have been of great interest since the Hawking-Page phase transition [3] between stable large black hole and thermal gas is explained as the confinement/deconfinement phase transition of gauge field [4] inspired by the AdS/CFT correspondence [5–7].After treating the cosmological constant as a pressure with its conjugate quantity being the thermodynamic volume in thermodynamic phase space of charged AdS black holes [8–14], the small/large black hole phase transition is established in [15], which is exactly analogous to the liquid/gas phase transition of the van der Waals fluid. This kind of black hole phase transitions has attracted much attention (see the recent review papers [16, 17]). Besides, this semiclassical method of analogue is generalized to study the microscopic structure of black holes and sheds some light on the black hole molecules [18] and microscopic origin of the black hole reentrant phase transition [19]. The study is also extended to the quantum statistic viewpoint, as the superfluid black holes are reported recently [20]. In Lovelock gravity with conformally coupled scalar field, the authors present the first example of λ phase transition, which is a line of (continuous) second-order phase transitions and was famous for the successful quantum and microscopic interpretation of superfluidity of liquid 4He.In this paper, we present another example of black holes containingλ phase transition in Horava gravity, which is a candidate of quantum gravity in ultrahigh energy [21]. The Horava-Lifshitz (HL) black hole solutions, thermodynamics, and phase transitions have attracted a lot of attention [22–29] (see [30] for a review on the recent development of various areas). The general dimensional HL black hole solutions are also introduced [31]. We will consider the extended thermodynamics of general dimensional HL AdS black holes. It is shown that only the one with spherical horizon in four and five dimensions has λ phase transition. Note that the first example of “superfluid” black holes always has a hyperbolic horizon [20]. Particularly, six-dimensional HL AdS black holes exhibit infinitely many critical points in P-ν plane and the divergent points for specific heat, for which they only contain the “normal” black hole phase and the “superfluid” black hole phase disappears due to the physical temperature constraint; therefore there is no similar phase transition. In more than six dimensions, there is no P-ν critical behavior. After identifying parameter ϵ as the ordering field instead of pressure and temperature, we study the critical phenomena in different planes of thermodynamical phase space. We also obtain the critical exponents, which are the same as the van der Waals fluid.The paper is structured as follows: in next section, we present the extended thermodynamics of generalized topological HL black holes. Then we studyP-V criticality in Section 3. We show the λ phase transition for four and five dimensions and the discussion for six dimensions in Sections 4 and 5, respectively. In Sections 6 and 7, we discuss the critical phenomena and calculate the critical exponents in different planes. In final section, some concluding remarks are given.


## 2. Extended Thermodynamics of Generalized HL Black Holes

In this section, we present the extended thermodynamics of generalized topological HL black holes in(d+1) dimensions (d≥3). We begin with the action of HL gravity at the z=3 UV fixed point, which can be reexpressed as [31](1)S=∫dtL0+1-ϵ2L1,L0≡∫ddxgN2κ2KijKij-λK2+κ28κW4ΛW1-dλd-2R-dΛW,L1≡∫ddxgNκ28κW411-dλ1-d4-λR2-1-dλRijRij,where the first two terms in the L0 are the kinetic actions, while the residue corresponds to the potential actions. Rij is the Ricci tensor, R is the Ricci scalar, and Kij is defined by Kij=(1/2N)(g˙ij-∇iNj-∇jNi), which are based on the ADM decomposition of the higher dimensional metric, i.e., dsd+12=-N2dt2+gij(dxi-Nidt)(dxj-Njdt). Here the lapse, shift, and d-metric N, Ni, and gij are all functions of t and xi, and a dot denotes a derivative with respect to t. There are five constant parameters in the action: ΛW, λ, ϵ,  κ, and κW. Λ=(d/2d-2)ΛW is the cosmological constant. κ and κW have their origin as the Newton constant and the speed of light. λ represents a dynamical coupling constant which is susceptible to quantum corrections. We will fix λ=1 in the following paper, only for which general relativity can be recovered in the large distance approximation. In addition, we will only consider the general values of ϵ in the region 0≤ϵ2≤1, as ϵ=0 corresponds to the so-called detailed-balance condition, and HL gravity with ϵ=1 returns back to general relativity.Action Eq. (1) admits arbitrary dimensional topological AdS black holes with the metric(2)ds2=-frdt2+dr2fr+r2dΩd-1,k2 and the horizon function [23, 31] (3)fr=k-2ΛW1-ϵ2r2d-1d-2-r2-d/2c01-ϵ2+ϵ21-ϵ224ΛW2rdd-12d-22,where dΩd-1,k2 denotes the line element of a (d-1)-dimensional manifold with constant scalar curvature (d-1)k, and k=0,±1 indicates different topology of the spatial space. c0 is integration constant, which is related to the black hole mass(4)M=-Ωd-1,k2c316πGN1d-2ΛWc0=-c0ΛW, where we have chosen the natural units and Ωd-1,k2=16(d-2)π.In AdS space-time, the cosmological constant is introduced as the thermodynamical pressure [15]:(5)P=-Λ8π=-d16πd-2ΛW. Then we can rewrite the horizon function as(6)fr=k+32πPr21-ϵ2dd-1-4r2-d/2d-2MPπd1-ϵ2+64ϵ2P2π2rd1-ϵ22d2d-12, and the mass is(7)M=64Pπr+dd-12d-2d+1-ϵ2dk2r+d-416Pd-2π+4kr+d-2d-2d-2, where r+ denotes the event horizon which is the largest positive root of f(r)=0. The conjugate thermodynamic volume of pressure is(8)V=64πr+dd-12d-2d-1-ϵ2dk2r+d-4P2d-2π. The entropy and temperature are presented in [31] with the following forms:(9)S=4πr+21+kd1-ϵ2ln⁡r+8d-2Pπr+2+S0,d=3,16πr+d-1d-12d-21+kdd-12d-21-ϵ232d-2d-3Pπr+2+S0,d≥4,(10)T=1024P2π2r+4+64kd-1d-2Pπr+2+k2dd-12d-41-ϵ28d-1πr+32πPr+2+kdd-11-ϵ2.It is easy to check the first law of thermodynamics:(11)dM=TdS+VdP, while the Smarr relation always fails to exist. This can be easily found because of the existence of S0 (and logarithmic term for d=3) in black hole entropy, which is not fixed and may be calculated by invoking the quantum theory of gravity as argued in [23]. Note that ϵ scales as [L]0, and there is no other dimensional quantity in extended thermodynamic phase space. In this meaning, the validity of the Smarr relation will bring another physical consideration on the parameter S0.In order to analyze the global thermodynamic stability and phase transition of the HL black hole, it is always to study the Gibbs free energy:(12)G=H-TS=M-TS, as the black hole mass M should be considered as the enthalpy H in the extended thermodynamic phase space. For the local thermodynamic stability, one can turn to the specific heat of black hole(13)C=dMdT=dM/dr+dT/dr+.We present the above two quantities in the Appendix, as their forms are very complicated.


## 3.P-V Criticality

According to (10), we can obtain the equation of state (EOS):(14)1024P2π2r+4+64kd-1d-2Pπr+2+k2dd-12d-41-ϵ2-8d-1πr+32Pπr+2+kdd-11-ϵ2T=0,which reflects the double-valuedness of the pressure in the extended thermodynamic phase space. On the other hand, we can derive the pressure P(T,r+) as(15)PT,r+=d-132πr+2-kd+2k+2πr+T+4k+2πr+T2-kdϵ24k-kd+8πr+T,while another one is the negative pressure branch. After taking a series expansion, it leads to(16)P=d-1T4r+-kd-1dϵ2+d-432πr+2+Or+-3. Comparing the above equation with the Van der Waals equation(17)P=Tν-b-aν2≃Tν+bTν2-aν2+Oν-3, one can easily find the specific volume ν∝r+. Therefore we will just use the horizon radius r+ in EOS instead of the specific volume ν and study the P-r+ behavior in the following paper.To consider theP-V criticality, we can focus on(18)∂P∂r+=0,∂2P∂r+2=0, to find the critical points. As the direct differentiation of P (see (15)) is too complicated, we prefer to employ the implicit differentiation on EOS (see (14)) and the above equations, and we can derive two simple conditions:(19)512P2πr+3+16d-1kd-2r+-6πr+2TP-kdd-121-ϵ2T=0,(20)96Pπr+2-12d-1πr+T+d-1d-2k=0. They lead to the critical points:(21)Pc=kd-112πX+2-d96πr+2,(22)Tc=kXr+, by which, the EOS (see (14)) is simplified as an equation of ϵ and results in the following condition:(23)ϵc=±22d-5Xπ+2d-73dd-4, where(24)X=3d-3±-3d-1d-512π.It is interesting that there is no critical volume or horizon radius, but the critical parameterϵ. To start the physical discussion, we should firstly calculate the physical critical points with real and positive pressure and temperature, which lead to the constraint (25)1≤d≤5.Namely, there is P-V criticality only in four, five, and six dimensions. Besides, the physical critical points are simplified as(26)Pc=k23γ-148πr+2,Tc=3kγ6πr+,ϵc=±299+63γ,d=3;Pc=k8πr+2,Tc=k2πr+,ϵc=±232,d=4;Pc=k8πr+2,Tc=k2πr+,ϵc=±255,d=5,where γ=±1. It is easy to find that positive pressure and temperature require the conditions (k=1,γ=1). In four dimensions (d=3), the critical behavior is studied in [29].Totally, we conclude that only four-, five-, and six-dimensional HL AdS black holes with spherical horizon have physical critical points, as shown in Table1.Table 1
The critical points of HL AdS spherical black holes.
Dimensions P c T c ϵc critical relationPcr+/Tc Four 2 3 - 1 48 π r + 2 3 6 π r + ±299+63 6 - 3 24 ≈ 0.178   Five 1 8 π r + 2 1 2 π r + ± 2 3 2 1 4   Six 1 8 π r + 2 1 2 π r + ± 2 5 5 14In next sections, we will give the physical discussion about the critical phenomena, i.e., the infinitely critical points and continuous second-order phase transitions. Especially for four-dimensional case, it is reported in [29], which is called “peculiar critical phenomena”. In Section 4, we will conclude that they are actually the famous λ phase transition after studying the specific heat CP of HL black holes. We also present the λ phase transition in five dimensions. In Section 5, it is found that six-dimensional HL AdS black holes only contain “normal” black hole phase and thus no similar phase transition. In Sections 6 and 7, we discuss the critical phenomena and calculate the critical exponents in different planes, by identifying parameter ϵ as the ordering field instead of pressure and temperature, respectively.


## 4.λ Phase Transition in Four and Five Dimensions

In this section, we discuss phase transitions in four and five dimensions. Back to the critical points in Table1, it is interesting that there is no critical volume or horizon radius, but the critical parameter ϵ. Moreover, the P-r+ oscillatory behavior and the classical “swallow tail” characterizing the first-order phase transition are controlled by the parameter ϵ, instead of the temperature T, which is clearly shown in Figure 1. The P-r+ diagrams are plotted at the same temperature T and G-T diagrams are plotted at the same pressure P. When ϵ>ϵc, there exist the small/large black holes phase transition in both four and five dimensions, which is exactly the same as the liquid/gas phase transition of van der Waals fluid. Especially for ϵ=ϵc, the P-r+ curves become a critical isotherm having an inflection point, and the second-order phase transition emerges.Figure 1
Curves ofP-r+ at the same temperature T=1/10 and G-T at the same pressure P=1/10 in four and five dimensions for different ϵ. When ϵ>ϵc, there are the P-r+ oscillatory behavior and the classical “swallow tail” characterizing the small/large black holes phase transition. When ϵ=ϵc, the second-order phase transition emerges and the dotted lines highlight the points where the second derivative of the Gibbs free energy G diverges.On the other hand, black holes with spherical horizon and arbitrary massM always have event horizon r+>0, which is arbitrary as well. As a result, four and five dimensional HL AdS black holes with ϵ=ϵc exhibit infinitely many critical points with arbitrary temperature Tc and horizon radius r+ as(27)Pc=23-1π4Tc2,Tc=36πr+,d=3;Pc=π2Tc2,Tc=12πr+,d=4.Namely, every isotherm in P-r+ diagrams is a critical isotherm, which has an inflection point at(28)r+=36πT,d=3;12πT,d=4.One can find it easily in Figure 2, and the dotted lines in P-r+ diagrams describe the position of all critical points (see (28)) at arbitrary temperature T. Besides, in G-T diagrams, there is no first-order phase transition but rather a line of second-order phase transitions, for which the phase transition points are highlighted by the dotted lines in Figure 2. The dashed lines in G-T diagrams describe the position of all phase transition points(29)T=4P23-1π,d=3;2Pπ,d=4at arbitrary pressure P, where the second derivative of the Gibbs free energy G diverges.Figure 2
Curves ofP-r+ at the different temperature T and G-T at the different pressure P in four and five dimensions for ϵ=ϵc. Every isotherm in P-r+ diagrams is a critical isotherm, and the dotted lines describe the position of all critical points. A line of second-order phase transitions is shown in G-T diagrams, and the dashed lines describe the position of all phase transition points.To study the continuous second-order phase transitions, we focus on the specific heatCP of black holes with ϵ=ϵc. From the CP-T diagrams in Figure 3, one can obtain that the specific heat CP always diverges at the critical temperature (see (29)). This is the classical λ line, i.e., the line of second-order phase transitions, in CP-T diagrams, which was famous in the discussion of superfluidity of liquid 4He. Similarly, in the P-T parameter space (right two plots of Figure 3), a line of critical points, i.e., the λ line, separates the two phases of system, which are the “superfluid” black hole phase and “normal” black hole phase. To determine the two phases, we consider the S-P diagrams, i.e., the middle two plots in Figure 3. The dashed lines highlight the critical pressure (see (27)). One can observe that the entropy of black holes with pressure smaller than the critical pressure is bigger, corresponding to the “normal” black hole phase. Another one is the “superfluid” black hole phase having smaller entropy and corresponding to black holes with pressure larger than the critical pressure. Particularly, the entropy of “superfluid” black hole phase is almost vanishing in five dimensions, while it seems to be negative in four dimensions. Note that the positivity of entropy depends on S0, which is not clear and could be fixed by counting microscopic degrees of freedom in quantum theory of gravity as argued in [23].Figure 3
Curves ofCP-T and S-P and P-T parameter space of black holes with ϵ=ϵc in four and five dimensions. The λ line in CP-T and P-T plots indicates the second-order phase transitions between “superfluid” black hole phase and “normal” black hole phase, which are distinguished in the S-P plots by the dashed lines.Finally, we conclude that this continuous second-order phase transition between small/large black holes corresponds to the phase transition between “superfluid” black hole and “normal” black hole, which is firstly reported in Lovelock gravity with conformally coupled scalar field [20].


## 5. “Normal” Black Hole Phase in Six Dimensions

In six dimensions, there are critical points inP-r+ plane as presented in Table 1. Actually, they describe infinitely many critical points with arbitrary temperature Tc and horizon radius r+ for HL black holes with ϵ=ϵc=±(2/5)5; i.e.,(30)Pc=π2Tc2,Tc=12πr+, which is exactly the same as the one in five dimensions. However, from Figure 4, it is strange that there is no P-r+ oscillatory behavior and classical “swallow tail” for different ϵ; especially for ϵ=ϵc, one can never find the second-order phase transitions. This is different from the cases in four and five dimensions, and six-dimensional black holes seem to have no phase transition.Figure 4
Curves ofP-r+ and G-T in six dimensions for different ϵ and ϵ=ϵc. There is no P-r+ oscillatory behavior and phase transition.To interpret the strange critical point, we could study the specific heatCP. One can look at the CP-T curve in the right plot of Figure 5. The specific heat CP diverges at the critical point (phase transition point)(31)T=2Pπ,while the dashed line highlighting a lower bound of temperature destroys the λ line. It is easy to calculate the minimal temperature of six-dimensional HL AdS spherical black holes with ϵ=ϵc, for which the temperature is reduced to(32)Tr+=64P2π2r+4+48Pπr+2+18πr+1+8Pπr+2. Considering the first-order derivative(33)T′=8Pπr+2-138πr+21+8Pπr+22, it leads to the minima at r+=1/8πP; i.e.,(34)Tmin=2Pπ,which is exactly the phase transition point (see (31)). This temperature bound could be treated as a physical temperature constraint, which cancels the “superfluid” black hole phase. This physical temperature constraint is equivalent to a upper bound of pressure, i.e., P≤πT2/2, for arbitrary temperature. Then in the S-P diagram in the middle one of Figure 5, one can also observe that there are no (“superfluid”) black holes for P>πT2/2, which correspond to the exact λ line in P-T parameter space as shown in the right plot in Figure 5. Therefore, even six-dimensional HL AdS black holes exhibit infinitely many critical points in P-ν plane and the divergent points for specific heat; there does not exist λ phase transition, as they only contain the “normal” black hole phase and the “superfluid” black hole phase disappears in the P-T parameter space due to the physical temperature constraint (see (34)).Figure 5
Curves ofCP-T and S-P and P-T parameter space of black holes with ϵ=ϵc in six dimensions. The dashed lines highlight the minimal temperature, which could be treated as a physical temperature constraint and cancel the “superfluid” black hole phase. Therefore there is no λ phase transition in six dimensions even when the specific heat diverges.


## 6. Critical Phenomena inϵ-r+ Plane

Because of the existence of infinite critical points, it is not able to calculate the critical exponents. Actually, to study the critical exponents forλ phase transitions of liquid He4, thermodynamic pressure is no longer the appropriate ordering field [32]. For the “superfluid” black hole, pressure should be instead of other parameters [20]. As for HL AdS black holes, there is only one option for the appropriate ordering field, i.e., parameter ϵ. Though ϵ is a dimensionless quantity, it does characterize the critical phenomena for four- and five-dimensional HL AdS black holes as shown later.Firstly, we study the critical points inϵ-r+ plane, for which the thermodynamic variables of EOS should be ϵ, r+(i.e., ν), T. Thus, we should firstly rewrite the EOS by rearranging the expression for temperature Eq. (10) for the chosen ordering field ϵ, which behaves as(35)ϵT,r+=±dd-1k+32πPr+2dd-12kd-1-32πPr+28πr+T-d-4k. We follow the conditions(36)∂ϵ∂r+=0,∂2ϵ∂r+2=0, to find the critical points. However, the direct differentiation of ϵ is too complicated; we also prefer to employ the implicit differentiation on EOS (see (14)) and the above equations. After a careful calculation, we obtain the same two conditions (19) and (20). Thus, one can easily get the critical point:(37)ϵc=±22d-5Xπ+2d-73dd-4,(38)rc=14kd-112πX+2-d6πP,(39)Tc=4kX6πPkd-112πX+2-d, where X has the value as (24) and P is a positive constant indicating AdS space-time.Following the discussions in Section3, it is shown that only four-, five-, and six-dimensional HL AdS black holes with spherical horizon have physical critical point, as shown in Table 2. Besides, in four and five dimensions, there exist the ϵ-r+ oscillatory behavior when T>Tc as shown in Figure 6 and the classical “swallow tail” characterizing the small/large black holes phase transition when ϵ>ϵc as shown in G-T diagrams of Figure 1. In six dimensions, it is easy to check that the critical point leads to ∂3ϵ/∂r+3=0. As a result, one can never find the ϵ-r+ oscillatory behavior as shown in Figure 6, and no first-order phase transition exists as shown in G-T diagrams of Figure 4.Table 2
The critical point of HL AdS spherical black holes inϵ-r+ plane.
Dimensions ϵc r c T c Four ± 2 9 9 + 6 3 ( 2 3 - 1 ) 48 π P 4 P ( 2 3 - 1 ) π   Five ± 2 3 2 1 8 π P 2 P π   Six ± 2 5 5 1 8 π P 2 P πFigure 6
Curves ofϵ-r+ with P=1/10 in four, five, and six dimensions for different temperature. In four and five dimensions, one can observe the ϵ-r+ oscillatory behavior when T>Tc, while no oscillatory behavior for six dimensions is observed.Finally, we consider the critical exponents in four and five dimensions. After introducing the dimensionless quantities(40)Ξ=ϵϵc,ω=r+rc-1,t=TTc-1, the ordering field can be expanded near the critical point as(41)Ξ=1+0.115t+0.234tω-0.042ω3+Otω2,ω4,d=3,1+14t+38tω-116ω3+Otω2,ω4,d=4,where the coefficients for four-dimensional case all have complicated forms, and their approximate values are given. The above expansions both have the same form as that for the van der Waals fluid and the RN-AdS black hole [15]. As the system can be characterized by the critical exponents(42)Cω∝t-α,ω∝tβ,κt=-ω-1∂ω∂Ξt∝t-γ,Ξ∝ωδ, one can obtain that(43)α=0,β=12,γ=1,δ=3. Particularly, ω∝t indicates that phase transition appears when T>Tc. Moreover, it is easy to check that they obey the scaling symmetry like the ordinary thermodynamic systems and in particular coincide with those for a superfluid [32].


## 7. Critical Phenomena inP-r+ Plane with “Temperature” ϵ

It is interesting to find that the parameterϵ controls the P-r+ oscillatory behavior instead of the temperature T, other than the pressure P, as shown in P-r+ diagrams of Figure 1. This indicates that there is still a critical phenomenon in P-r+ plane. For this case, pressure could still be treated as the ordering field, while ϵ should be considered as “temperature”. This is different from that for the first “superfluid” black holes [20].To study the corresponding critical phenomena, we can still follow the procedure studyingP-r+ criticality in Section 3, which leads to the same three critical equations: (14), (19), and (20). As the thermodynamic variables of EOS for this case should be P, r+ (i.e., ν), ϵ, we can find the following critical point:(44)Pc=d-112πX+2-d96πkX2T2,(45)rc=kXT, with the critical “temperature”(46)ϵc=±22d-5Xπ+2d-73dd-4, where X takes the value as (24) and T is a positive constant.As similar as the discussions in Section3, one can find that only four-, five-, and six-dimensional HL AdS black holes with spherical horizon have physical critical point, as shown in Table 3. In four and five dimensions, when “temperature” ϵ>ϵc, there exist the P-r+ oscillatory behavior (Figure 1) and the classical “swallow tail” characterizing the small/large black holes phase transition as shown in G-P diagrams of Figure 7. As for the case of six dimensions, it is easy to find that the critical point leads to ∂3P/∂r+3=0. Therefore, one can not observe the P-r+ oscillatory behavior as shown in Figure 4, and no first-order phase transition exists as shown in Figure 7.Table 3
The critical point of HL AdS spherical black holes inP-r+ plane with “temperature” ϵ.
Dimensions P c r c ϵ c Four ( 2 3 - 1 ) π T 2 4 3 6 π T ± 2 9 9 + 6 3   Five π T 2 2 1 2 π T ± 2 3 2   Six π T 2 2 1 2 π T ± 2 5 5Figure 7
Curves ofG-P with T=1/10 in four, five, and six dimensions for different “temperature” ϵ. In four and five dimensions, one can observe the classical “swallow tail” characterizing the small/large black holes phase transition when “temperature” ϵ>ϵc, while no “swallow tail” exists for six dimensions.Then, we can also calculate the critical exponents in four and five dimensions near the critical point. Similarly, we begin with the following dimensionless quantities:(47)p=PPc,ω=r+rc-1,τ=ϵϵc-1. The EOS can be reduced to the dimensionless case, for which its Taylor series expansion at the critical point takes the following forms:(48)p=1-563+9411τ+103+18τω-3-1ω3+Oτω2,ω4,d=3,1-8τ+12τω-12ω3+Oτω2,ω4,d=4.They still both have the same form as that for the van der Waals fluid and the RN-AdS black hole [15]. After introducing the critical exponents(49)Cω∝τ-α,ω∝τβ,κτ=-ω-1∂ω∂pτ∝τ-γ,p∝ωδ, it is easy to find(50)α=0,β=12,γ=1,δ=3, which satisfy the scaling laws of the ordinary thermodynamic systems. Besides, ω∝τ indicates that phase transition appears when ϵ>ϵc.


## 8. Discussion

In this paper, we study the extended thermodynamics of general dimensional HL AdS black holes and present another example of “superfluid” black holes. It is found that only four- and five-dimensional HL AdS black holes with spherical horizon have theλ phase transition, which correspond to the phase transition between “superfluid” black hole and “normal” black hole. After considering the behavior of entropy, the “superfluid” black hole phase and “normal” black hole phase are distinguished. Particularly, six-dimensional HL AdS black holes exhibit infinitely many critical points in P-ν plane and the divergent points for specific heat, for which they only contain the “normal” black hole phase and the “superfluid” black hole phase disappears due to the physical temperature constraint; therefore there is no similar phase transition. In more than six dimensions, there is no P-V critical behavior. After identifying parameter ϵ as the ordering field instead of pressure and temperature, we study the critical phenomena in different planes of thermodynamical phase space. We also obtain the critical exponents in both planes, which are the same as the van der Waals fluid.The “superfluid” black hole is firstly reported in Lovelock gravity with conformally coupled scalar field [20], which contains at least four free parameters in the action of theory. Comparing with it, the one in Horava gravity has only one free parameter ϵ. It is interesting to consider the necessary and sufficient conditions for general “superfluid” black hole; i.e., a black hole EOS must satisfy to display similar λ phase transition, which is still unclear. Besides, could the “superfluid” black holes appear in Einstein gravity? These are all interesting and left to be the future tasks.



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