Eur. Phys. J. C (2017) 77:423 DOI 10.1140/epjc/s10052-017-4989-0

Regular Article - Theoretical Physics

Thermodynamics, phase transitions and Ruppeiner geometry for Einstein–dilaton–Lifshitz black holes in the presence of Maxwell and Born–Infeld electrodynamics

M. Kord Zangeneh1,2,3,a, A. Dehyadegari2, M. R. Mehdizadeh4,5,b, B. Wang1,c, A. Sheykhi2,5,d 1 Department of Physics and Astronomy, Center of Astronomy and Astrophysics, Shanghai Jiao Tong University, Shanghai 200240, China 2 Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran 3 Physics Department, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz 61357-43135, Iran 4 Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran 5 Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

Received: 28 May 2017 / Accepted: 11 June 2017 © The Author(s) 2017. This article is an open access publication

Abstract In this paper, we first obtain the higher-dimen- sition points, there is no thermally stable black hole between sional dilatonÐLifshitz black hole solutions in the presence these two points and we have discontinuous small/large black of BornÐInfeld (BI) electrodynamics. We find that there are hole phase transitions. As expected, for small black holes, two different solutions for the cases of z = n+1 and z = n+1 we observe finite magnitude for the Ruppeiner invariant, where z is the dynamical critical exponent and n is the number which shows the finite correlation between possible black of spatial dimensions. Calculating the conserved and thermo- hole molecules, while for large black holes, the correlation dynamical quantities, we show that the first law of thermody- is very small. Finally, we study the Ruppeiner geometry and namics is satisfied for both cases. Then we turn to the study thermal stability of BI charged Lifshtiz black holes for dif- of different phase transitions for our Lifshitz black holes. We ferent values of z. We observe that small black holes are ther- start with the HawkingÐPage phase transition and explore mally unstable in some situations. Also, the behavior of the the effects of different parameters of our model on it for correlation between possible black hole molecules for large both linearly and BI charged cases. After that, we discuss black holes is the same as for the linearly charged case. In the phase transitions inside the black holes. We present the both the linearly and the BI charged cases, for some choices improved Davies quantities and prove that the phase transi- of the parameters, the black hole system behaves like a Van tion points shown by them are coincident with the Ruppeiner der Waals gas near the transition point. ones. We show that the zero temperature phase transitions are transitions in the radiance properties of black holes by 1 Introduction using the LandauÐLifshitz theory of thermodynamic fluctu- ations. Next, we turn to the study of the Ruppeiner geome- It has been over 40 years since Bekenstein and Hawking first try (thermodynamic geometry) for our solutions. We investi- disclosed that black hole can be considered as a thermody- gate thermal stability, interaction type of possible black hole namic system, with characteristic temperature and entropy molecules and phase transitions of our solutions for linearly [1Ð4]. Taking into account the fact that black holes have no and BI charged cases separately. For the linearly charged hair, there are no classical degrees of freedom to account case, we show that there are no phase transitions at finite for such thermodynamic properties. It is a general belief that ≥ < temperature for the case z 2. For z 2, it is found that the thermodynamic properties of a system may reflect the statis- number of finite temperature phase transition points depends tical mechanics of underlying relevant microscopic degrees on the value of the black hole charge and there are not more of freedom. But the detailed nature of these microscopic than two. When we have two finite temperature phase tran- gravitational states has remained a mystery. The BekensteinÐ Hawking entropy, S = A/(4hG¯ ), depends on both Planck’s a e-mail: [email protected] constant and Newtonian gravitational constant, implying b e-mail: [email protected] that the thermodynamics of black holes may relate quan- c e-mail: [email protected] tum mechanics and gravity. Recently, there has been some d e-mail: [email protected] progress in understanding the microscopic degrees of free- 123 423 Page 2 of 21 Eur. Phys. J. C (2017) 77:423 dom of the black hole entropy, for example in string the- standing of the black hole phase transition [30,31]. They ory [5Ð7] as well as loop quantum gravity [8Ð10]. But the found that some second moments in the fluctuation of rele- accounts of the black hole entropy are not complete and they vant thermodynamic quantities diverge when the black hole only work within some particular models and some special becomes extreme. This divergence shows that the thermody- domains where string theory and loop quantum gravity can namic fluctuation is tremendous and the rigorous meaning apply. Besides, despite counting very different states, many of the thermodynamical quantities is broken down. This is inequivalent approaches to quantum gravity obtain identical exactly the characteristic of the thermodynamic phase tran- results and it is not clear why any counting of microstates sition point. At this phase transition point, the Hawking tem- should reproduce the same BekensteinÐHawking entropy perature is zero which indicates that for the extreme black [11]. The statistical mechanical description of the black hole hole there is only super-radiation but no Hawking radiation, entropy is still not elegant. which is in sharp difference from that of the non-extreme On the other side, black hole can be heated or cooled black holes. Black holes phase transition in the context of through absorption and evaporation processes. According to LandauÐLifshitz theory have been investigated in [32,33]. Boltzmann’s insight, if a system can be heated, it must have Recently, further differences in dynamical properties before microscopic structures. Recently, in [12], possible micro- and after the black hole thermodynamical phase transition has scopic structures of a charged anti-de Sitter black hole have been disclosed in [34Ð36]. Also, in [37Ð42], black hole phase been studied and some kind of interactions between possi- transitions have been studied from holographic point of view. ble micromolecules have been investigated by an interest- A question now arises: how we can further understand this ing physical tool, the Ruppeiner geometry. Derived from the macroscopic thermodynamic phase transition in black hole thermodynamic fluctuation theory, the Ruppeiner geometry physics? for example whether there is a microscopic explana- [13,14] is considered a powerful tool in exploring the possi- tion of this thermodynamic phase transition. The Ruppeiner ble interactions between black hole microscopic structures. geometry is a possible tool we can use to investigate the The sign of the Ruppeiner invariant R (the Ricci scalar of thermodynamic phase transitions from microscopic point of the Ruppeiner geometry) was argued to be useful for identi- view. This method is safer to determine true phase transi- fying the physical systems similar to the Fermi (Bose) ideal tions than other methods since, regardless of the microscopic gas when R > 0(R < 0) or the classical ideal gas when model, R has a unique status in identifying microscopic order R = 0[15]. Besides, the sign of the Ruppeiner invariant R (which is at the foundation of phase transitions at microscopic can further be used to interpret the type of dominant interac- level) from thermodynamics [20,21]. Some attempts in this tion between molecules of a thermodynamic system. When direction have been reported in [43Ð54]. In a recent work R > 0, there is a repulsive interaction between molecules, [46], it was found that the divergence of the Ruppeiner invari- when R < 0 the interaction is attractive, and for R = 0 there ant coincides with the critical point in the phase transition in a is no interaction in the microstates [16Ð18]. Moreover, the holographic superconductor model. It is interesting to inves- magnitude of the Ruppeiner invariant |R| measures the aver- tigate whether the Ruppeiner geometry [20,21] can present age number of correlated Planck areas on the event horizon us further reason to determine which of the thermodynamical for a black hole system [19]. For a review of the descrip- discussions mentioned above is valid for describing the ther- tion of the Ruppeiner geometry in black hole systems, we modynamical phase transition. In particular, we would like to refer to [20,21] and the references therein. Further studies of explore whether the Davies phase transition conjecture can molecular interactions of black holes, based on the Ruppeiner reflect some special properties in microstructures and be in geometry, have been carried out in [12,22,23]. consistence with the Ruppeiner geometry description. If the The phase transition is another interesting topic in black Davies conjecture does not have the microscopic explana- holes thermodynamics. Davies discussed the thermodynamic tion, we will further think about how to improve the Davies phase transition of the black holes by looking at the behav- conjecture to describe the black hole phase transition. ior of the heat capacity [24Ð26]. He claimed that the dis- We will employ the black hole in Lifshitz spacetime as continuity of the heat capacity marks a second order phase a configuration to study our physical problems mentioned transition in black holes. However, it was argued that phys- above. This spacetime was first introduced in [55], which ical properties do not show any particularity at this discon- respects the anisotropic conformal transformation t → λzt, tinuity point if compared with other heat capacity values; x → λx, where z is dynamical critical exponent. For the Lif- for example the regularity of the event horizon is not lost shitz spacetime, it is necessary to include some matter sources and the black hole internal state remains uninfluenced [27]. such as massive gauge fields [56Ð60] or higher-curvature cor- Thus, it is hard to accept the discontinuity point of the heat rections [61] to guarantee the asymptotic behavior of the capacity as a true physical point of the phase transition. Lifshitz black hole. It is difficult to find an analytic Lifshitz Employing the LandauÐLifshitz theory of thermodynamic black hole solution for arbitrary z, although some attempts fluctuations [28,29], Pavon and Rubi gave a deep under- have been performed [62]. This makes the discussion of ther- 123 Eur. Phys. J. C (2017) 77:423 Page 3 of 21 423 modynamics for such a black hole difficult. Fortunately, in descriptions of the phase transition such as the Davies conjec- EinsteinÐdilaton gravity with a massless gauge field, it is ture and the LandauÐLifshitz method. We try to give more possible to find an exact Lifshitz black solution for arbitrary microscopic understanding of the thermodynamical phase z [63,64]. This model is suggested in the low energy limit transitions in the black hole system. We explore the thermo- of string theory [65]. While thermodynamical behaviors of dynamic geometry (Ruppeiner geometry) for linearly and uncharged and charged EinsteinÐdilatonÐLifshitz black holes nonlinearly charged Lifshitz solutions separately and show have been revealed in [63,66] and [64], respectively, ther- the properties of interactions between possible black hole modynamics of uncharged GaussÐBonnetÐdilatonÐLifshitz molecules. To the best of our knowledge, there is no study of solution has been studied in [67]. It is also interesting to study thermodynamic geometry on Lifshitz solutions in the litera- Lifshitz black hole solutions in the presence of other gauge ture. Interestingly enough, by studying Ruppeiner geometry, fields such as the power-law Maxwell field [68], the logarith- we have found that our solutions show the Van der Waals like mic [69] and exponential [70] nonlinear electrodynamics. For behavior near critical point in some cases. example, thermodynamics and thermal and dynamical stabil- The layout of the paper is as follows. In the next sec- ities of EinsteinÐdilatonÐLifshitz solutions in the presence of tion, we give the basic field equations and obtain the BI power-law Maxwell field have been studied in [71]. In the charged LifshitzÐdilaton black hole solutions. In Sect. 3,we context of AdS/CFT [72Ð74] application, the electrical con- first explore the satisfaction of the first law of thermody- ductivity were explored for exponentially [75] and logarith- namics for LifshitzÐdilaton black holes in the presence of BI mic [76] charged Lifshitz solutions. In the present work, we electrodynamics. Then we study different phase transitions shall consider the BornÐInfeld (BI) nonlinear electrodynam- including the HawkingÐPage phase transition and phase tran- ics in the context of EinsteinÐdilatonÐLifshitz black holes. sition at zero temperature for linearly and BI charged cases. The motivation for considering the BI-dilaton action comes In Sect. 4, we investigate thermodynamic geometry of the from the fact that the dynamics of D-branes and some soli- obtained solutions for linearly and nonlinearly BI charged ton solutions of supergravity are governed by the BI action cases by adopting the Ruppeiner approach. We finish with a [77Ð82]. Besides, the low energy limit of open superstring summary and closing remarks in Sect. 5. theory suggest the BI electrodynamic action to be coupled to a dilaton field [77Ð79]. It is surprising that many years 2 Action and asymptotic Lifshitz solutions before the appearance of the BI action in superstring theory, in the 1930s, this nonlinear electrodynamics was introduced In this section, we intend to obtain exact (n +1)-dimensional for the first time, with the aim of solving the infinite self- dilatonÐLifshitz black holes in the presence of BI nonlin- energy problem of a point-like charged particle by imposing ear electrodynamics. Our ansatz for the line elements of the a maximum strength for the electromagnetic field [83]. spacetime is [64,88] In this paper, we will first look for a general (n + 1)- 2z ( ) 2 2 dimensional Lifshitz black hole solution in the context of 2 r f r 2 l dr 2 2 ds =− dt + + r d − , (1) EinsteinÐdilaton gravity in the presence of BI electrodynam- l2z r 2 f (r) n 1 ics. We will show that the general metric function has dif- ferent solutions for z = n + 1 and z = n + 1 cases. It is where z(≥ 1) is the dynamical critical exponent and important to note that the difference in the metric function n−1 i−1 has not been observed in the previous studies on LifshitzÐ     d2 = dθ 2 + dθ 2 sin2 θ dilaton black holes [71,75,76]. Based on this general solu- n−1 1 i j i=2 j=1 tion, we will study the thermodynamics of LifshitzÐdilaton black holes coupled to a linear Maxwell field and BI nonlin- is an (n − 1)-dimensional hypersurface with constant curva- ear electrodynamics. We will show that the HawkingÐPage ture (n − 1)(n − 2) and volume ωn−1.Asr →∞, the line phase transition [84] exists both in the presence of linear and elements ( 1) reduce asymptotically to the Lifshitz spacetime, nonlinear electrodynamics. There are some attempts to study 2z 2 2 2 phase transitions of uncharged Lifshitz solutions for fixed z 2 r dt l dr 2 2 ds =− + + r d − . (2) [85] or in three [86] and four [87] dimensions. The HawkingÐ l2z r 2 n 1 Page phase transition revealed in this paper is interesting, since it depends on different values of z in different space- On the other side, as pointed out above, we would like to time dimensions in the presence of linear Maxwell and non- consider BI nonlinear electrodynamics. In the absence of the linear BI electrodynamic fields. We will further concentrate dilaton field, the BI Lagrangian density is written as [83] our attention on understanding the thermodynamic phase    transition from microstructures. We shall examine the rela- F L = 4β2 1 − 1 + , (3) tion between the Ruppeiner geometry and thermodynamical 2β2 123 423 Page 4 of 21 Eur. Phys. J. C (2017) 77:423

β 2 where is the BornÐInfeld  parameter related to the Regge n − 1  λ   μν 2 i −4λi /(n−1) slope α as β = 1/ 2πα . F = Fμν F is Maxwell invari- ∇ + L + e Hi = 0, (8) 8 2 ant in which Fμν = ∂[μ Aν] where Aμ is electromagnetic   i=1 μν potential. One of the effects of the presence of a dilaton field μ L F = 0, (9)  F  is its coupling with the electromagnetic field. Thus, in the −4λ /(n−1) μν μ e i (H ) = 0, (10) presence of the dilaton field we deal with a modified form i for the BI Lagrangian density including its coupling with the where we use the convention XY = ∂ X/∂Y . Using the metric dilaton scalar field [89,90], ansatz (1), electromagnetic field equations (9) and (10) can ⎛  ⎞ be solved immediately as −8λ /(n−1) λ /( − ) e F L(F, )= 4β2e4 n 1 ⎝1 − 1 + ⎠ , (4) 4λ /(n−1) z−n β2 qβe r 2 F = , rt ϒ (11) − λ /( − ) ( ) = z n 4 i n 1 , λ Hi rt qi r e (12) where is a constant. The Lagrangian density of the string-  generated EinsteinÐdilaton model [65] with two Maxwell where ϒ = 1 + q2l2z−2/(β2r 2n−2), and (r) can be gauge fields [64] in the presence of BI electrodynamics can obtained by subtracting (tt) and (rr) components of Eq. (7 ) be written in an Einstein frame as and solving the resulting equation. We find 1 4 L = R − (∇ )2 − 2 √   16π n − 1 (n − 1) z − 1 r  (r) = ln . (13) 2 2 b −4 λi /(n−1) − e Hi + L(F, ) , (5) i=1 Substituting Eqs. (11), (12) and (13) in the field equations (7) and (8), one can solve the equations for f (r) to obtain

⎧ − −  ⎪ m (n−2)2l2 4β2l2b2z 2 4β2l2b2z 2 n−z ⎪ 1 − + − + + − − + − ϒr dr, for z = n + 1, ⎨ rn z 1 (n+z−3)2r2 r2z 2(n−1)(n−z+1) (n−1)rn z 1 f (r) = ⎪    (14) ⎪ ( − )2 2 β2 2n 2 +ϒ ⎩ 1 − m + n 2 l − 4 b l 1 − ϒ + ln 1 , for z = n + 1, r2n 4(n−1)2r2 (n−1)2r2n 2

where we should set where R is the Ricci scalar and and the λi are some μν constants. In Lagrangian (5), Hi = (Hi )μν (Hi ) where √ n − n − λ =− − ,λ = √ 1 ,λ = √ 2 , = ∂ z 1 1 2 (Hi )μν [μ (Bi )ν] and (Bi )μ is gauge potential. In the z − 1 z − 1 β L large limit, recovers the EinsteinÐdilatonÐMaxwell − (z − 1) b2(n−1) (n − 1)(n − 2)(z − 1)b2(n−2) q2 = , q2 = , Lagrangian in its leading order [64,71] 1 (z + n − 2) l2(z−1) 2 2(z + n − 3)l2(z−1) − λ /( − ) lim 16πL = ···−e 4 n 1 F (n + z − 1)(n + z − 2) β→∞ =− , (15)   2l2 e−12λ /(n−1) F2 1 + + O . (6) so that the field equations are fully satisfied. In the solution β2 β4 8 (14), m is a constant which is related to the total mass of  √ n+1 Varying the action S = M d x −gL with respect to the black brane as we will see in next section. The integral of ( ) = + metric gμν, the dilaton field and electromagnetic potentials the last term of f r for z n 1 can be done in terms of ( ) Aμ and (Bi )μ leads to the following field equations: hypergeometric function. Thus, f r can be written as

gμν m (n − 2)2l2 4b2z−2l2β2(1 − ϒ) Rμν − 2 + 2L F F − L(F, ) f (r) = 1 − + + n − 1 r n+z−1 (n + z − 3)2r 2 (n − 1)(n − z + 1)r 2z−2  2 2z−2 2z 2 4q b l ϒ − λ /( − ) + − 4 i n 1 ( + − )( − + ) 2(n+z−2) e Hi n z 3 n z 1 r  = 2n + z − 4 3n + z − 5 i 1 × F , , , − ϒ2 . 1 − − 1 (16) 4 λ 2n 2 2n 2 − ∂μ ∂ν + 2L F Fμλ Fν n − 1 Note that solution (16) obviously satisfies the fact that 2 −4λ /(n−1) λ f (r) → 1asr →∞(note that F (a, b, c, 0) = 1). The − 2 e i (H )μλ (H ) = 0, (7) i i ν ( ) β i=1 behavior of f r for large is 123 Eur. Phys. J. C (2017) 77:423 Page 5 of 21 423

⎧   − − − ⎪ m (n−2)2l2 2q2b2z 2l2z q4b2z 2l4z 2 1 ⎪ 1 − + − + + + − − + − + O , for z = n + 1, ⎨ rn z 1 (n+z−3)2r2 (n−1)(n+z−3)r2n 2z 4 4(n−1)(3n+z−5)β2r4n 2z 6 β4 ( ) = f r ⎪   (17) ⎪ ( − )2 2 2 2n 2n+2 4 2n 4n+2 ⎩ 1 − m + n 2 l + q b l − q b l + O 1 , for z = n + 1. r2n 4(n−1)2r2 (n−1)2r4n−2 8(n−1)2β2r6n−4 β4

z 2 1−z which reproduces the result of [71] for every z in linear (n + z − 1)r+ (n − 2) l T = + π z+1 2−z Maxwell case. The behaviors of the metric function for 4 l 4π(n + z − 3)r+ = + = +   z n 1 and z n 1 have been depicted in Fig. 1a q2lz−1b2z−2 q4l3z−3b2z−2 1 − + + O . and b, respectively. It is notable to mention that in the case of 2n+z−4 4n+z−6 2 β4 2π(n − 1)r+ 8π(n − 1)r+ β z = n + 1, there is no Schwartzschild-like black hole since (20) in this case f (r) goes to positive infinity as r goes to zero. However, for z = n + 1, we may have Schwartzschild-like The entropy of the black holes can be calculated by using black hole (dash-dotted line in Fig. 1a) in addition to non- the area law of the entropy [2,91,92] which is applied to extreme (solid line) and extreme (dotted line) black holes almost all kinds of black holes in Einstein gravity including and naked singularity (dashed line). For non-extreme case, dilaton black holes [93Ð96]. Therefore, the entropy of the there are two inner (Cauchy) and outer (event) horizons. In black brane per unit volume ωn−1 becomes both Fig. 1a and b, we see that the larger the nonlinearity β n−1 parameter is, the smaller the distance between two inner r+ β S = . (21) and outer horizons is so that, for large enough ,wehave 4 just one horizon (extreme case) or naked singularities. The Schwartzschild-like case occurs for lower β’s in the case of Having Eqs. (11), (13) and (15) at hand, we can find electro- z = n + 1asFig.1ashows. magnetic gauge potential At = Frtdr in terms of hyperge- As one can see in (17), the fourth term in expansions ometric function as for both z = n + 1 and z = n + 1 cases reproduce the   qb2z−2 1 n + z − 3 3n + z − 5 charge term of [71] in linear Maxwell case as one expects. A (r) =− F , , , 1 − ϒ2 . t ( + − ) n+z−3 − − The temperature of the black hole horizon can be obtained n z 3 r 2 2n 2 2n 2 via (22)     The large β behavior of gauge potential is in agreement with 1 1 b a T = − ∇bχa∇ χ  , (18) 2π 2  [71] r=r+   qb2z−2 q3b2z−2l2z−2 1 χ = ∂ At (r) =− + + O . where t is the Killing vector and r+ is the radius of (n + z − 3)r n+z−3 (3n + z − 5) r 3n+z−5β2 β4 event horizon. Using (18), one can calculate the Hawking (23) temperature as  r z+1 f   In next section, we will study thermodynamics of dila- T =  z+1  ton Lifshitz black holes in the presence of BI electrodynam- 4πl = r r+ ics by seeking for satisfaction of thermodynamics first law (n + z − 1)r z ( − )2 1−z = + + n 2 l through calculation of conserved and thermodynamic quan- π z+1 2−z 4 l 4π(n + z − 3)r+ tities. We also show that our Lifshitz solutions can exhibit the 2 2z−2 2−z HawkingÐPage phase transition. Then we discuss the inside β b r+ (1 − ϒ+) + , (19) phase transitions of our Lifshitz black holes. π(n − 1)lz−1 where prime denotes the derivative with respect to r and ϒ = ϒ ( = ) + r r+ . For the temperature one has the same 3 Thermodynamics of Lifshitz black holes formula as (19) for the two cases z = n + 1 and z = n + 1. β One can check that, for large ,(19) reduces to the tempera- 3.1 First law of thermodynamics ture of EinsteinÐMaxwellÐdilatonÐLifshitz black holes [71], namely This subsection is devoted to a study of the first law of ther- modynamics for LifshitzÐdilaton black hole solutions in the

123 423 Page 6 of 21 Eur. Phys. J. C (2017) 77:423

(a) (b)

Fig. 1 The behavior of f (r) versus r for l = 1, b = 0.8andq = 1.3

σ B 0 presence of BI nonlinear electrodynamics. As the first step, where is the determinant of the boundary metric, Kab we calculate the fundamental quantity for thermodynamics is the background extrinsic curvature, na is the timelike unit discussions namely mass. For this purpose, we apply the normal vector to the boundary B and ξ b is a timelike Killing modified subtraction method of Brown and York (BY) [97Ð vector field on the boundary surface. Applying the above 99]. In order to use this method, the metric should be written modified BY formalism, the mass of the space-time per unit in the form volume ωn−1 can be calculated as

2 2 2 dR 2 2 ds =−X(R)dt + + R d − . (24) (n − 1)m Y (R) n 1 M = , (28) 16πlz+1 For our case, it is clear that R = r and thus where the mass parameter m can be obtained from the fact that f (r+) = 0as

⎧ 2 2 n+z−3 − ⎪ n+z−1 + (n−2) l r+ + 4b2z 2l2β2(1−ϒ+) ⎪ r+ ( + − )2 z−n−1 ⎪ n z 3  (n−1)(n−z+1)r+  = + , ⎨⎪ 2 2z−2 2z for z n 1 + 4q b l ϒ+ , 2n+z−4 , 3n+z−5 , − ϒ2 n+z−3 F 1 − − 1 + m(r+) = (n+z−3)(n−z+1)r+ 2n 2 2n 2 (29) ⎪ ⎪    ⎪ 2 2 2n−2 ⎩⎪ (n−2) l r+ 4β2b2nl2 1+ϒ+ r 2n + − 1 − ϒ+ + ln for z = n + 1. + 4(n−1)2 (n−1)2 2

Now, we turn to a calculation of the electric charge of the r(R)2z f (r(R)) r(R)2 f (r(R)) X(R) = , Y (R) = . (25) solution. Using the Gauss law, we can calculate the electric l2z l2 charge via The metric of background is chosen to be the Lifshitz metric  (24) i.e. 1 n−1 μ ν Q = r L F Fμνn u d, (30) 4π r(R)2z r(R)2 X0(R) = , Y0(R) = . (26) l2z l2 where

z The quasilocal conserved mass can be obtained through μ 1 l  n = √ dt = √ dt, √ − z ( ) 1 2 gtt r f r M = d ϕ σ (Kab − Khab) √ 8π B ν 1 r f (r)    u = √ dr = dr, − 0 − 0 0 aξ b, grr l Kab K hab n (27) are, respectively, the unit spacelike and timelike normals to the hypersurface of radius r.Using(30), the charge per unit 123 Eur. Phys. J. C (2017) 77:423 Page 7 of 21 423 volume ωn−1 can be computed as perature. In addition, as one can see from Fig. 2, there are some other choices of the parameters that show a non-zero − qlz 1 positive minimum for temperature T . The influences of Q = . (31) min 4π different parameters on Tmin can be seen from Fig. 2. When we increase the dimension n, Tmin increases too, while it The electrostatic potential difference (U) between the hori- decreases with increasing z. Comparing Fig. 2a and b, one zon and infinity is defined as finds that the effect of nonlinearity implies increasing in Tmin.   μ  μ The behaviors illustrated in Fig. 2 show a HawkingÐPage U = Aμχ →∞ − Aμχ . (32) r r=r+ phase transition for the obtained solutions. Let us have a closer look on Fig. 2. In the first part of the T ÐS curves Using Eqs. (22) and (32), one can obtain the electric potential n−1 where we have small black holes (note that S = r+ /4),   ∂ /∂ < 2z−2 + − + − T S 0, which implies negative heat capacity and there- = qb 1 , n z 3 , 3n z 5 , − ϒ2 , U + − F 1 + fore small black holes are thermally unstable. But in the large (n + z − 3)r n z 3 2 2n − 2 2n − 2 + black hole part of the curves we have a positive heat capac- (33) ity and therefore large black holes are thermally stable. In addition to small and large black holes, we have a thermal which is the same for the cases z = n + 1 and z = n + 1. In Lifshitz or radiation solution too. Since small black holes are order to investigate the first law of black hole thermodynam- thermally unstable, the system has a choice between large ics, we should obtain the Smarr-type formula for mass (28). black hole and thermal Lifshitz solutions and chooses one With Eqs. (29), (31) and (21) at hand, the mass can be writ- of them according to the Gibbs free energy. The Gibbs free ten as a function of the extensive thermodynamic quantities energy, S and Q in the form of

⎧ ( − + )/( − ) ⎪ (n−1)(4S)(n+z−1)/(n−1) (n−1)(n−2)2(4S)(n+z−3)/(n−1) β2(4S) n z 1 n 1 (1−) ⎪ + + − + − ( − ) ⎪ 16πlz 1 16πlz 1(n+z−3)2 4πlz 1b2 1z (n−z+1) ⎨⎪ − − for z = n + 1, + 4(n−1)π Q2b2z 2l1 z  , 2n+z−4 , 3n+z−5 , − 2 , ( , ) = ( + − )( − + )( )(n+z−3)/(n−1) F 1 2n−2 2n−2 1 M S Q ⎪ n z 3 n z 1 4S (34) ⎪ ⎪ /( − )    ⎩ (n−1)(4S)2n n 1 (n−2)2 S2 β2b2n 1+ + + − 1 −  + ln , for z = n + 1, 16πln 2 4π(n−1)ln 4π(n−1)ln 2

  = + π 2 2/(β2 2) where 1 Q S . Calculations show that G (T, U) = M − TS− QU, (37) intensive quantities     can be obtained by using (19), (21), (28), (31) and (33). Fig- ∂ M ∂ M T = U = , ures 3 and 4 show the behavior of the Gibbs free energy ∂ and ∂ (35) S Q Q S for some choices of the parameters. The two up and bottom branches correspond to small and large black holes, respec- coincide with those computed by Eqs. (19) and (33). There- tively. The positive Gibbs free energy shows that the system fore, the thermodynamics quantities satisfy the first law of is in a radiation phase, while there is a HawkingÐPage phase thermodynamics, transition at the intersection point of the bottom branch and G = 0. This fact that the Gibbs free energy of large black dM = T dS + UdQ, (36) holes always has the lower energy in comparison to small ones confirms the above arguments as regards their thermal for the two solutions z = n + 1 and z = n + 1. stability. As one moves rightward on the temperature axis In the next part of this section, we will discuss the in the GÐT diagram, first one has the radiation regime or HawkingÐPage and inside black hole phase transitions for the thermal Lifshitz solution for which G > 0. At G = 0, our Lifshitz solutions. the HawkingÐPage phase transition between the thermal Lif- shitz case and large black holes occurs, and for G < 0weare 3.2 Black hole phase transitions at the large black hole phase. The temperature at which the phase transition occurs is called the HawkingÐPage temper- 3.2.1 Hawking–Page phase transition ature THP. The effects of a change in the electric potential U, the critical exponent z and the nonlinearity parameter β can As it is clear from Fig. 1, there are some parameter choices for be seen from Figs. 3 and 4. Increasing the electric potential U which we have extreme black holes and therefore zero tem- 123 423 Page 8 of 21 Eur. Phys. J. C (2017) 77:423

(a) (b)

Fig. 2 The behavior of T versus S for l = 1, b = 1

(a) (b)

Fig. 3 The behavior of G versus T for linearly charged case with l = 1, b = 1andn = 3

(a) (b)

Fig. 4 The behavior of G versus T for nonlinearly charged case with l = 1, b = 1andn = 3

and the critical exponent z makes THP lower. Also, the lower on the microscopic viewpoint. The two macroscopic ways the nonlinearity parameter β is, the lower the HawkingÐPage are Davies [24] and LandauÐLifshitz [28,29] methods, which temperature THP is. Note that a lower β makes the electro- discuss, respectively, the behavior of heat capacities and ther- dynamics more affected by nonlinearity. modynamic fluctuations. Thermodynamic geometry or Rup- peiner geometry [16,20,21] is the microscopic way; it dis- cusses the phase transitions in addition to type and strength of 3.2.2 Phase transitions inside the black hole interactions. In what follows, we discuss the relation between the phase transitions predicted by Ruppeiner geometry and There are at least three well-known ways to discuss the phase the Davies method. Next, we will turn to the LandauÐLifshitz transitions inside the black hole. Two of these ways are based theory of thermodynamic fluctuations. on the macroscopic point of view and one of them is based 123 Eur. Phys. J. C (2017) 77:423 Page 9 of 21 423

Ruppeiner and Davies phase transitions dicted by the Ruppeiner invariant. These quantities are the specific heat at constant electrical potential, CU , the ana- In order to discuss thermodynamic geometry, one should log of the volume expansion coefficient, α, and the ana- study the divergences, sign and magnitude of Ricci scalar log of the isothermal compressibility coefficient κT defined corresponding to Ruppeiner metric (usually called Ruppeiner as invariant) to determine phase transitions and strength and       type of dominated interaction between possible black hole ∂ S 1 ∂ Q 1 ∂ Q C = T α = ,κ= molecules [16,20,21]. To do that, we define the Ruppeiner U ∂ , ∂ T ∂ . T U Q T U Q U T metric in (M, Q) space where the entropy S is the thermo- (41) dynamic potential,

2 As one can see in the Appendix, these thermodynamic quan- ∂ S α gαβ =− , X = (M, Q). (38) tities have the forms ∂ X α∂ X β  M 1 M ∂T  The above metric can also be rewritten in the Weinhold form, C = T SS ,α=− SQ ,κ=−α  . (42) U M M T ∂  HS,Q Q HS,Q U Q 2 1 ∂ M α gαβ = , Y = (S, Q). (39) T ∂Y α∂Y β It is obvious that these quantities show the same phase transitions as the Ruppeiner geometry because all of them The Ruppeiner invariant corresponding to (39) can be M diverge at roots of HS,Q and CU vanishes at zero tempera- expressed in a general form as ture where R diverges. To show the coincidence of the Rup- peiner phase transitions and C divergences, some proofs N(S, Q) U R = , have been presented in [50,54]. The above quantities can be D( , ) (40) S Q considered as improved Davies quantities [24] which show the phase transitions to coincide with the Ruppeiner ones. In where N and D stand for the numerator and the denomi- R the next part, we study the LandauÐLifshitz theory of ther- nator of . The divergences of the Ruppeiner invariant is 2 modynamic fluctuations to explore the possible signature of D M determined by the roots of , equal to T HS,Q where black hole phase transitions and the properties of black hole M = − 2 radiance. HS,Q MSSMQQ MSQ is the determinant of the Hes- 2 sian matrix and XYZ = ∂ X/∂Y ∂ Z. Of course, at these divergence points the numerator N should be finite. These Landau–Lifshitz theory (non-extreme/extreme phase transi- M divergences show both zero temperature and vanishing HS,Q . tion) M The root of HS,Q may show the boundary between thermal stability and instability. For thermal stability, in addition to Here, we seek any possible effect of a transition on black positivity of the determinant of the Hessian, MQQ and MSS hole radiance by using the LandauÐLifshitz theory of ther- should be positive too [100,101]. modynamic fluctuations [28,29]. We focus on the (3 + 1)- It is remarkable that at the point where MSS vanishes or dimensional linearly charged case. The extension to higher- equivalently heat capacity at constant charge CQ diverges, dimensional or nonlinearly charged cases is trivial and gives we have a thermally unstable system due to negativity of no novel result. Based on LandauÐLifshitz theory [28,29], ˙ M = in a fluctuation–dissipative process, the flux Xi of a given HS,Q if MSQ 0 (which occurs in many black hole sys- tems). Thus, the heat capacity at constant charge CQ can- thermodynamic quantity Xi is given by not be a suitable thermodynamic quantity to show the phase  ˙ transition of such systems when we have two changing ther- Xi =− ijχ j , (43) modynamic parameters, for instance S and Q. There is some j work in the literature (for instance [102]) in which the cor- rectness of the Ruppeiner method for recognizing the phase where a dot shows the temporal derivative and χi and ij transitions has been judged by comparing the Ruppeiner and are, respectively, the thermodynamic force conjugate to the ˙ CQ transition points. This procedure of course seems to be flux Xi and the phenomenological transport coefficients. incorrect according to what we pointed out above. Also, as In addition, the rate of entropy production is expressed we discussed in the introduction, the divergences of R are by safer if we determine phase transitions. On the other hand,  in [43,48], the authors have suggested some suitable ther- ˙ ˙ S = ±χi Xi , (44) modynamic quantities to show the phase transitions pre- i 123 423 Page 10 of 21 Eur. Phys. J. C (2017) 77:423 where “+”(“−”) holds for the entropy rate contributions where which come from the non-concave (concave) parts of S.The π z−1 z/2+1 π 2 2z−2 second moments corresponding to the flux fluctuations are χ = zl S χ = b QS. M z−4 and Q z−2 (we set kB = 1) 2  4  !   ˙ ˙ δXi δX j = ij +  ji δij, (45) The mass loss rate is given by [103] where the mean value with respect to the steady state is dM =− ασ 4 + dQ . ˙ b T U (49) denoted by the angular brackets and the fluctuations δXi are dt dt the! spontaneous deviations from the value of steady state ˙ The first term on the right side of Eq. (49) is the thermal Xi . To guarantee that the correlations are zero when two mass loss corresponding to Hawking radiation, which is just fluxes are independent, the Kronecker δij is put in Eq. (45). the StefanÐBoltzmann law, with b = π 2/15 (we set h¯ = According to [71], the mass M, electric potential energy 1) as the radiation constant. The constant α depends on the U and temperature T can be obtained for (3+1)-dimensional number of species of massless particles and the quantity σ is linearly charged case as the cross-section of geometrical optics. The second term on (4S)(z+2)/2 (4S)z/2 2π Q2b2z−2 the right side of Eq. (49) is responsible for the loss of mass M = + + , (46) corresponding to charged particles. In fact, it is the UdQ 8πlz+1 8πz2lz−1 zlz−1(4S)z/2 term which arises in the first law of black hole mechanics. With references to what was explained and computed πb2z−2 Q 2z−4 U = and T = , (47) above, one can calculate the second moments or correlation z−2 z−1 z/2 π z−1 z/2+1 z2 l S zl S functions of the thermodynamical quantities, where ! 2z−3 ! 22z−5b2−2z δM˙ δM˙ =− M˙ , δQ˙ δQ˙ = Q˙ , z z+1 −2 2−z 2 2z−2 2 π z−1 z/2+1 π 2  = S + 4z(z + 2)S l − 4 zπ b Q . ! zl S! SQ δM˙ δQ˙ = U δQ˙ δQ˙ , (50) We know that in extreme black hole case, the Hawking tem- perature on the event horizon vanishes and therefore in this " ! π 2z2l2z−2 Sz+2 ! π 2b4z−4 Q2 casewehave = 0. Using Eq. (46), we can obtain the δS˙δS˙ = δM˙ δM˙ + 4z−42 4z−2z2l2z−2 Sz entropy production rate as # ! πb2z−2 Q ! × δQ˙ δQ˙ − δM˙ δQ˙ S˙(M, Q) = χ M˙ − χ Q˙ , (48) 2z−3zlz−1 Sz/2 M Q " # πzlz−1 Sz/2+1 πb2z−2 Q =− M˙ + Q˙ , (51) 2z−5 z2z−2lz−1 Sz/2

  2 ! (z − 2)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z(z + 2)42−zb2z−2 Q2 δT˙ δT˙ = 22 ! 4S δM˙ δM˙   2 42−zπ 2b4z−4 Q2 (z − 1)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z42−zb2z−2 Q2 + 2 2z−2 z+22 ! z l S δQ˙ δQ˙    πb2z−2 Q (z − 1)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z42−zb2z−2 Q2 (z − 2)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z(z + 2)42−zb2z−2 Q2 − z−2 z−1 z/2+22 ! z2 l S δM˙ δQ˙ z−5 $ 2 − + =− (z − 2)Sz + 4z2(z + 2)l 2 Sz 1 πzlz−1 Sz/2+3 − − 2 + π 2z(z + 2)42 zb2z 2 Q2 M˙  πb2z−2 Q − (z − )Sz + z2(z + )l−2 Sz+1 z−4 z−1 z/2 1 4 2 2 zl S  % − − − − + π 2z42 zb2z 2 Q2 Sz − 16π 24 z Q2b2z 2z(z + 1) Q˙ , (52)

123 Eur. Phys. J. C (2017) 77:423 Page 11 of 21 423

  ! πzSz/2 (z − 2)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z(z + 2)42−zb2z−2 Q2 δ ˙δ ˙ = S T z− z− ! 2 3l 12 δM˙ δM˙   π 3b4z−4 Q2 (z − 1)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z42−zb2z−2 Q2 + 3z−8 z−1 z/22 ! z2 l S δQ˙ δQ˙   π 2b2z−2 Q (3z − 4)Sz + 12z2(z + 2)l−2 Sz+1 + π 2z(z + 4)42−zb2z−2 Q2 − − ! 22z 52 δM˙ δQ˙   (z − 2)Sz + 4z2(z + 2)l−2 Sz+1 + π 2z(z + 2)42−zb2z−2 Q2 =− 2z−2  " # l S πb2z−2lz−1 Q × M˙ + Q˙ . (53) − 1 z 2z 2zS2 ! ! ˙ ˙ ˙ ˙ It is! clear that the second moments δSδS , δT δT and Combining Eqs. (55) and (56), we arrive at δS˙δT˙ diverge for the extreme black hole case where  & ' 3 4z−4 (z+2)/(z+1) z+1 dQ 16e b Q z r+ vanishes (see Eq. 47). It means that there is a phase tran- =−  − , , (57) z/(z+1) dt ( + ) 3z−3 z + 1 QQ0 sition in this case. This phase transition is between extreme z 1 l Q0 and non-extreme black holes for which we have a sudden  , change in emission properties. In the non-extreme case, the where [a b] is incomplete gamma function. When r+ black hole can give off particles and radiation through both Q,Eq.(57) reduces to spontaneous Hawking emission and superradiant scattering,   4 6z−6 3 z+1 dQ 64e b Q r+ whereas in the extreme case, the black hole can just radiate ≈− exp − +··· , (58) 2 4z−4 2z+1 via superradiant scattering. dt (z + 1)m l r+ QQ0 As one can see from Eq. (49), M˙ and Q˙ are related. There- fore, all of the above second moments can be reexpressed in where we have used terms of Q˙ . Let us calculate Q˙ for our case. The rate of charge " # z /( + ) loss can be stated as  − , x ≈ exp (−x) x1 z 1 z + 1    "   # dQ ∞ 2π π √ 1 1 − = −  θ φ , × + O +··· , e g d d dr (54) 2 3 (59) dt r+ 0 0 x x −1 where  is the rate of electronÐpositron pair creation per four- in which x 1. volume and e is charge of electron. According to Schwinger’s In the following section, we turn to the study of thermody- theory [104]for(3+ 1) dimensions, the rate of electronÐ namic geometry of our black hole solutions to figure out the positron pair creation in a constant electric field E is behavior of the black hole’s possible molecules and phase transitions.  "   # 4e2b4z−4 1 e3 E  = E2 − + O +··· , 2z−2 exp 1 2 πl EQ0 m 4 Ruppeiner geometry (55)

2z−2 2 z−1 In this section we study thermodynamic geometry of the where Q0 = 4πeb /πm l and m is the mass of elec- LifshitzÐdilaton black holes for linearly Maxwell and non- tron. In the presence of linear Maxwell electrodynamics, the + linearly BI gauge fields, separately. We have introduced this electric field is E = Q/r z 1 and therefore   method in Sect. 3.2.2 with focus on the study of the phase z− z+ 4e2b4 4 Q2 r 1 transitions which occur at divergence of the Ruppeiner invari-  = exp − πl2z−2r 2z+2 QQ ant R. In addition to divergences, R has other properties "   0 # e3 Q which give us information about thermodynamic of the sys- × 1 + O +··· . (56) R m2r z+1 tem. The sign of gives us the information about the dom- inated interaction between possible black hole molecules, 123 423 Page 12 of 21 Eur. Phys. J. C (2017) 77:423 while its magnitude measures the average number of corre- where " lated Planck areas on the event horizon [16,19Ð21]. R > 0 2[1+(z−2)/(n−1)] means the domination of repulsive interaction, R < 0shows F(S, Q) ≡ S the attraction dominated regime and when R vanishes the z(n + z − 3)(n + z − 1)S2[1+(z−1)/(n−1)] system behaves like an ideal gas i.e. there is no interaction. + −4/(n−1)( − )( − )2 2 In the following, we first study thermodynamic geometry in 2 z 2 n 2 l ( − + )/( − ) # the presence of linear Maxwell electrodynamics. Then we π 2(n + z − 3)Q22 n 4z 7 n 1 − . (64) extend our study to nonlinearly charged black holes where (n − 1)(n − 2)2b2(1−z) BI electrodynamics has been employed. There is just a nec- The numerator N of (40) is a complicated finite function essary comment. As we stated before in Sect. 3.2.2, for ther- of S and Q in this case, including long terms that we do not mal stability, M , M and HM = M M − M2 QQ SS S,Q SS QQ SQ express explicitly for brevity. However, as mentioned in Sect. should be positive [100,101]. One can show that the positiv- D M 3.2.2, one can find the denominator in the form of ity of H , and MQQ (MSS) imposes the positivity of MSS S Q  (M ). Therefore, we just turn to the study of the signs of 2 QQ D( , ) = M , M S Q TLM HS,Q (65) HS,Q and MQQ in the following discussions to guarantee the thermal stability. M where TLM and HS,Q have been give in (60) and (63), respec- tively. 4.1 Linear Maxwell case Having Eqs. (63) and (65) at hand, we are in a position to investigate the divergences of R, which play the central The mass and Hawking temperature of black holes in the role in thermodynamic geometry discussions, and also ther- presence of linear Maxwell (LM) electrodynamics are mal stability of system. As one can see from Eqs. (63) and z 2 1−z (65), for z = 2, the divergences occur just in the case of the (n + z − 1)r+ (n − 2) l TLM = + extremal black holes where T = 0. For z = 2, in addition 4πlz+1 π( + − ) 2−z LM 4 n z 3 r+ to extremal black hole case, R diverges in zeros of (63). In − − q2lz 1b2z 2 the latter case, we can calculate the corresponding tempera- − , (60) 2n+z−4 F = 2π(n − 1)r+ ture by solving 0forQ and then putting this Q in Eq. (60) to arrive at

( − + )/( − ) /( − ) (n+z− )/(n− ) ( + − ) 2z n 1 n 1 z n 1 (n − 1)(4S) 1 1 T = n z 1 2 S . MLM(S, Q) = z+1 (66) 16πlz+1 π(2 − z)l (n − 1)(n − 2)2(4S)(n+z−3)/(n−1) + The above temperature is negative for z > 2 i.e. there is no π( + − )2 z−1 16 n z 3 l black hole at this diverging point and therefore the diver- − ( − − )/( − ) 2π Q2b2z 2(4S) 3 n z n 1 R + . (61) gences of occur just for extremal black hole case when (n + z − 3)lz−1 z > 2. However, for z < 2 when T > 0, we can see an upper limit in entropy and charge of system. The largest entropy S As we mentioned above, for investigating thermal stability for which F = 0 (which we call it critical entropy S ) can be M c we need to check the signs of MQQ and HS,Q. In our case calculated by finding the extremum point where ∂F/∂ S = 0 as π 2z−2 −(n+z−3)/(n−1) b S 4/(n−1) MQQ = > 0. (62) /( −n) z(n + z − 1)(n + z − 2)2 (n + z − 3) lz−122(z−2)/(n−1) S2 1 = , (67) c (2 − z)(n − 2)2l2

Thus, in order to disclose the thermal stability of system, at which we need to study the sign of the determinant of the Hessian ( + − ) (n − 2)2 n z 2 matrix. We find Q2 = c l−2(n+z−3)b2(z−1)π 2 ⎧ (n − 1)(n + z − 2)2−n−z(n + z − 1)3−n−z 2 2z−2 −2[2+(z−2)/(n−1)] × ⎪ (z−2)(n−2) b S 5 ⎨⎪ − F(S, Q) z = 2 2 (n + z − 3) 4(n−1)(n+z−3)2l2z 2   M = , n+z−3 HS,Q ⎪ 2 ⎪ ( + ) 2 (n−5)/(1−n) 2(n−2)/(1−n) × − 1 (68) ⎩ n 1 b 2 S z = 2 z (n−1)2l4 (63) and 123 Eur. Phys. J. C (2017) 77:423 Page 13 of 21 423

(a) (b)

M = = Fig. 5 The behavior of HS,Q versus T for the linear Maxwell case with l b 1

(a) (b)

Fig. 6 The behavior of the Ruppeiner invariant R versus T for the linear Maxwell case with l = b = 1

 2  (n−2z+3)/(n−1) ∂ F (n + z − 3) 4 We have summarized the above discussion in Figs. 6 and  =− ∂ S2 ( − )2 7. These figures also show the sign of the Ruppeiner invari- S=Sc n 1 " # − ant for different choices of the parameters that determines the (n + z − 1)(n + z − 2) z 2 z × < 0. (69) type of interaction between black hole molecules [16,20,21]. ( − )( − )2 2 2 z n 2 l Figure 6a is depicted for RN-AdS case (n = 3, z = 1). In One should note that the absolute value of Qc is also the this case, it can be seen that, for Q > Qc, the Ruppeiner largest charge value which satisfies F = 0. Another remark invariant diverges only for extremal black holes. As Fig. 6a to be mentioned is that (67) imposes an upper limit on the size shows, there is also a range of T for which R < 0, namely of black hole too (see (21)). At this point, the corresponding the dominated interaction between black hole molecules is temperature can be obtained: attractive. Furthermore, the interactions near zero tempera- ture is the same as interactions of Fermi gas molecules near  ( − )/ ( − )z + − 2 z 2 zero temperature [16]. According to Fig. 5a, for Q > Qc, =  n 2 n z 1 . Tc (70) HM is positive (also M > 0 (see (62))), and therefore 2πl (z(n + z − 2))z 2 − z S,Q QQ the system is stable for all T region. For Q = Qc, in addition to the zero temperature, we have another temperature (Tc), For charges greater than Qc, the Ruppeiner invariant diverges where a divergence of R occurs (see Fig. 6a). At zero tem- only in the case of extremal black holes. For Q = Qc,in perature, the Ruppeiner invariant goes to +∞, while at Tc it addition to TLM = 0, we have one other divergence in R goes to −∞. The latter case is similar to the Van der Waals specified by (67) and (70). For Q < Qc, in addition to TLM = 0, we have at most two other divergences, since the order gas phase transition at the critical point in the sense that at R −∞ of the polynomial in terms of S is always lower than 3 for the phase transition temperature, goes to [14,21]. For = R n ≥ 3 and z < 2. One should note that, in the latter case, the Q Qc, becomes positive when we go away from the second divergence (Tc) on the temperature axis. In Q = Qc temperature region between two divergences is not allowed M M < case, HS,Q is positive and just vanishes at Tc (Fig. 5a), so, since HS,Q 0 (Fig. 5). 123 423 Page 14 of 21 Eur. Phys. J. C (2017) 77:423

(a) (b)

Fig. 7 The behavior of the Ruppeiner invariant R versus T for the linear Maxwell case with l = b = 1

the system is always thermally stable. For Q < Qc, there these phase transitions can be considered as small/large black are three divergences; one at T = 0, one at T < Tc and one holes phase transitions. The first reason for this argument is > M at T Tc. In this case, according to Fig. 5a, HS,Q is neg- that as temperature increases, entropy or equivalently size ∂ /∂ = −1 > ative in the temperature region between two roots and show of black hole increases (note that S T MSS 0). instability. This not-allowed region is equivalent to the tem- Therefore,on the left side of phase transition points where perature region between two divergences of the Ruppeiner the temperature is lower, we have small size black holes and invariant for Q < Qc (Fig. 6a). Figures 5b and 6b show the the right side where the temperature is higher we have large same properties for black holes with different parameters. In size ones. This fact can also be seen from the behavior of this case, Tc is greater than one of previous case while Qc is the Ruppeiner invariant magnitude at the two sides of the lower. Figure 7 shows the behavior of the Ruppeiner invari- phase transitions. For small black holes, we expect a finite ant for z ≥ 2. As this figure shows, there are just divergences correlation between possible black hole molecules (of course at T = 0. The properties of black hole molecular interac- far from phase transition points) because those are close to tions (R > 0: Repulsion, R = 0: No interaction and R < 0: each other. For large black holes, we expect the correlation Attraction) depend on parameters such as the dimension of between possible molecules to tend to a small value near zero space time and the charge, in this case. According to Eq. (63), since molecules become approximately free. These expected = M > for z 2, HS,Q is always positive. For z 2, we can find behaviors can be seen in Fig. 6. = M Q from TLM 0 and put it in HS,Q to obtain

4.2 BornÐInfeld case (n + z − 1)b2z−22(n−5)/(1−n) HM = > . S,Q,T =0 z (n− )/(n− ) 0 (71) (n − 1)(n + z − 3)l2 S 2 1 For the BornÐInfeld case, we can calculate the Ruppeiner invariant by using Eqs. (19), (34) and (39). The Ruppeiner M > Thus, since HS,Q nowhere vanishes for z 2 (see discus- invariant in this case is very complicated due to the presence sions below (66)) and is positive at T = 0 according to above of hypergeometric functions. Therefore, in this case we dis- equation, it is positive throughout the temperature region and cuss the thermodynamic geometry non-analytically and by therefore the system is always thermally stable for z > 2. looking at plots. We study the cases z < 2, z > 2, z = 2 and Regarding the nature of the phase transition occurring at z = n + 1 separately. First, we study the case z < 2. Figure zero temperature where the Ruppeiner invariant diverges, we 8a shows that changing β can cause a change in the dominant discussed in previous section via LandauÐLifshitz theory of interaction. For instance, in a range of T ,wehavenegative thermodynamic fluctuations. However, regarding the phase R (attraction) for β = 1 (note that in this range the system is transitions occurred at divergences of R at finite tempera- thermally stable as one can see from Figs. 9a and 10a). For tures, we can give some comments here. We have seen two β = 1, the system behaves like a Fermi gas at zero tempera- kinds of phase transitions here for z < 2(seeFig.6) namely ture, namely R goes to positive infinity at zero temperature continuous (for Q = Qc where R diverges at just one finite [16]. For β = 0.82, the Ruppeiner invariant diverges at two temperature or entropy) and discontinuous (for Q < Qc points; one of them is the zero temperature. According to where R diverges at two finite temperatures or entropies Fig. 9a, for temperatures lower than the second divergence M and we have a jump between these two points since there point, HS,Q is negative and therefore the system is thermally is no thermally stable black hole between them). Both of unstable. Since MQQ > 0 (Fig. 10a), the system is thermally 123 Eur. Phys. J. C (2017) 77:423 Page 15 of 21 423

(a) (b)

Fig. 8 The behavior of the Ruppeiner invariant R versus T for the BornÐInfeld case with l = b = 1

(a) (b)

M = = Fig. 9 The behavior of the determinant of the Hessian matrix HS,Q versus T for the BornÐInfeld case with l b 1

(a) (b)

Fig. 10 The behavior of MQQ versus T for BornÐInfeld case with l = b = 1 stable just for temperatures greater than the temperature of like Van der Waals gas at phase transition temperature i.e. second divergence for β = 0.82. Figure 8a shows that there R goes to negative infinity at this point [14,21]. For z > 2, is no extremal black hole for β = 0.5 i.e. we have a black the behavior of R is depicted in Fig. 11a. It can be seen that hole with just a single horizon. The allowed temperatures are the type of dominated interaction changes for different β and greater than the temperature of divergence according to Figs. we have negative R for some cases. In this case, we have a 9a and 10a. In Figs. 8b, 9b and 10b, respectively, the Rup- behavior like a Fermi gas at zero temperature for extremal M β = . peiner invariant, HS,Q and MQQ are depicted for different black holes. For 0 13, there is a divergence at non-zero choices of the parameters. It is remarkable that, in the case of temperature; for lower temperatures, the system is unstable β = . = M 0 046, Fig. 8b shows that the behavior of system looks (Fig. 11b, c). In the case of z 2, HS,Q and MQQ are

123 423 Page 16 of 21 Eur. Phys. J. C (2017) 77:423

(a)

(b) (c)

R M = . = . = = Fig. 11 The behavior of the Ruppeiner invariant , HS,Q and MQQ versus T for the BornÐInfeld case with b 0 9, l 0 76, n 6, z 3and Q = 0.018

ent dominated interaction. For this case, possible molecules of black hole behave like Fermi gas at zero temperature. The last case is z = n + 1. In this case MQQ is

 2n 2  b β ( − 1) MQQ = , (73) z=n+1 4π(n − 1)Q2ln

which is positive for all temperatures. The behavior of the M Ruppeiner invariant and HS,Q are depicted in Fig. 13a and b for this case, respectively. As one can see the type of inter- action is β-dependent for some temperatures. For β = 0.04, M HS,Q is positive just for temperatures greater than the finite temperature of divergence (Fig. 13b) and therefore the sys- Fig. 12 The behavior of the Ruppeiner invariant R versus T for the tem is thermally stable for this range of temperatures. BornÐInfeld case with z = 2andl = b = 1 Most of the phase transitions discovered above in the  presence of BI electrodynamics at finite temperatures can-  (n + 1)b2 M  = , not be interpreted as small/large black hole phase transitions HS,Q ( − )/( − ) /( − ) (72) z=2 2 n 5 n 1 l4(n − 1)2 S2n n 1  because in these cases small size black holes are unstable. Further studies to disclose the nature of these phase transi- and tions are called for.  2  b π MQQ = , z=2 (n − 1)lS 5 Summary and closing remarks which are always positive and therefore the system is always stable and R experiences no divergence (Fig. 12). In this case, In many condensed matter systems, fixed points governing for different values of nonlinear parameter β,wehavediffer- the phase transitions respect dynamical scaling, t → λzt, 123 Eur. Phys. J. C (2017) 77:423 Page 17 of 21 423

(a) (b)

R M = . = . = = Fig. 13 The behavior of the Ruppeiner invariant and HS,Q versus T for the BornÐInfeld case with b 0 91, l 0 72, n 3, z 4and Q = 0.012 x → λx where z is the dynamical critical exponent. The geometry (Ruppeiner invariant) at finite temperature for the gravity duals of such systems are Lifshitz black holes. In this case z ≥ 2. For z < 2, it was found that the number of paper, we first sought for the (n+1)-dimensional BornÐInfeld divergences (which show the phase transitions) at finite tem- (BI) charged Lifshitz black hole solutions in the context of peratures depend on the value of the charge Q. We introduced dilaton gravity. We found that these solutions are different a critical value for the charge, Qc; for greater values there is for the cases z = n + 1 and z = n + 1. We found that both no divergence at finite temperature, for values lower than it solutions and showed that the solution for the case z = n + 1 there are at most two divergences and for Q = Qc, there is can never be Schwartzschild-like. Then we studied the ther- just one diverging point for the Ruppeiner invariant. For the modynamics of both cases by calculating conserved and ther- case of Q < Qc, there is a thermally unstable region for sys- modynamical quantities and checking the satisfaction of the tems between two divergences at finite temperatures. So, this first law of thermodynamics. After that, we looked for the phase transition can be claimed as a discontinuous phase tran- HawkingÐPage phase transition for our solutions, both in the sition between small and large black holes. For small black cases of linearly and BI charged black holes. We studied holes not close to the transition point, we observed a finite this phenomenon and the effects of different parameters on magnitude for the Ruppeiner invariant R. This is reasonable it by presenting the behaviors of temperature T with respect since the magnitude of R shows the correlation of possible to entropy S at fixed electrical potential energy U and also black hole molecules. Also, for large black holes the magni- Gibbs free energy G with respect to T . Then we turned to tude of the Ruppeiner invariant tends to a very small value discuss the phase transitions inside the black holes. In this as expected. For Q = Qc, the solutions show a continuous part, we first presented the improved Davies quantities that small/large black holes phase transition at finite temperature. show the phase transition points coincided with the ones of In the case of BI charged solutions, we investigated the Rup- the Ruppeiner geometry. This coincidence has been proved peiner geometry and thermal stability for z < 2, z > 2, z = 2 directly in the Appendix. All of our solutions, provided that and z = n +1 separately. In some of these cases, small black those are thermally stable at zero temperature, show the diver- holes were thermally unstable. So, more studies are called for gence at this point both from the Ruppeiner and the Davies to discover the nature of phase transitions at diverging points points of view. Using the LandauÐLifshitz theory of thermo- of R. In both the linearly and nonlinearly charged cases, dynamic fluctuations, we showed that this phase transition is for some choices of the parameters, the black hole system a transition of the radiance properties of black holes. At zero behaves like a Van der Waals gas near the transition point. temperature, an extreme black hole can just radiate through Finally, we would like to suggest some related interesting superradiant scattering whereas a non-extreme black hole at issues which can be considered for future studies. It is inter- finite temperature can give off particles and radiation via both esting to repeat the studies such as regards the HawkingÐPage spontaneous Hawking radiation and superradiant scattering. phase transition, Ruppeiner geometry and LandauÐLifshitz Next, we turned to study Ruppeiner geometry for our solu- theory for black branes to discover the effect of different tions. We investigated thermal stability, interaction type of constant curvatures of the (n − 1)-dimensional hypersurface possible black hole molecules and phase transitions of our on those phenomena. One can also seek for any signature solutions for linearly and nonlinearly BI charged cases sep- of different phase transitions discovered here such as the arately. For the linearly charged case, we showed that there HawkingÐPage phase transition, and phase transitions deter- are no diverging points for Ricci scalar of the Ruppeiner mined by the Ruppeiner geometry, in dynamical properties

123 423 Page 18 of 21 Eur. Phys. J. C (2017) 77:423 of solutions by investigating quasi-normal modes. Some of With the above relations at hand, one can show that      this work is in progress by the authors. ∂ ( , ( , )) ∂  ∂  ∂  ∂  T S Q U S  = T  − T  Q  U  ∂  ∂  ∂  ∂  ∂  Acknowledgements We are grateful to Prof. M. Khorrami for very S U S Q Q S U S S Q   useful discussions. MKZ would like to thank Shanghai Jiao Tong Uni-  ∂T  ∂U  versity for the warm hospitality during his visit. Also, MKZ, AD and ∂  ∂  ∂ = T  − Q S  S Q AS thank the research council of Shiraz University. This work has been ∂  ∂  S Q U  financially supported by the Research Institute for Astronomy & Astro- ∂ Q physics of Maragha (RIAAM), Iran.   S  ∂T  ∂U  ∂T  ∂U  ∂ ∂  − ∂  ∂ Open Access This article is distributed under the terms of the Creative = S Q Q S  Q S S Q Commons Attribution 4.0 International License (http://creativecomm ∂U  ∂  ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, Q S and reproduction in any medium, provided you give appropriate credit 2 M MQQMSS − M H , to the original author(s) and the source, provide a link to the Creative = SQ = S Q . (A4) Commons license, and indicate if changes were made. MSS MSS Funded by SCOAP3. In the last line of (A4), we have used (35). Equation (A4) M = − 2 shows that HS,Q MSSMQQ MSQ is in denominator of = (∂ /∂ ) CU T S T U and therefore it exactly diverges at the point where the Ruppeiner invariant diverges. To show this Appendix A: Suitable thermodynamic quantities to deter- fact for α, we should obtain mine phase transitions     ∂ ( , ( , )) ∂  ∂  ∂  T Q S U Q  = T  + T  S  . In [43,48], the authors have shown that the divergences of ∂  ∂  ∂  ∂  (A5) Q U Q S S Q Q U the specific heat at constant electrical potential, CU , the ana- log of the volume expansion coefficient, α, and the analog of As is well known the isothermal compressibility coefficient κT are in coinci- dence with the phase transitions specified by the Ruppeiner    ∂  ∂  ∂  invariant. The definitions of these quantities are S  =− S  U  , ∂  ∂  ∂  (A6) Q U U Q Q S       ∂ S 1 ∂ Q 1 ∂ Q and therefore we have C = T α = κ = U ∂ , ∂ and T ∂ . T U Q T U Q U T      ∂ ( , ( , )) ∂  ∂  ∂  ∂  (A1) T Q S U Q  = T  − T  S  U  ∂  ∂  ∂  ∂  ∂  Q U Q S S Q U Q Q S    ∂T  ∂U  ∂  ∂ ∂  Here we will prove that these quantities are exactly suitable T  S Q Q S =  −  to characterize the phase transitions shown by the Ruppeiner ∂ Q ∂U   S ∂ S Q  invariant. We showed in Sect. 4 that the divergences of the     ∂T  ∂U  − ∂T  ∂U  Ruppeiner invariant occurs at roots of the determinant of the ∂ Q ∂ S Q ∂ S Q ∂ Q M = − 2 = S  S Hessian matrix HS,Q MSSMQQ MSQ and also zero ∂U  M ∂ S Q temperature. In our proof, we will show that HS,Q exactly M M − M2 HM exists at denominator for all above suitable thermodynamic =− QQ SS SQ =− S,Q . quantities. MSQ MSQ Let us start with CU .Wehave (A7) α = −1 (∂ /∂ )     The above relation shows that Q Q T U diverges ∂ ( , ( , )) ∂  ∂  ∂  at the point where Ruppeiner invariant does. Finally, to obtain T S Q U S  = T  + T  Q  . ∂  ∂  ∂  ∂  (A2) a similar result for κ , we write S U S Q Q S S U T     ∂  ∂  ∂  ∂  U  =− U  T  =− 1 U  . ∂  ∂  ∂  α ∂  (A8) On the other hand we know that Q T T Q Q U Q T Q

κ = −1 (∂ /∂ )    The above relation shows that T Q Q U T is pro- ∂  ∂  ∂  α Q  =− Q  U  . portional to and therefore diverges at the same points as ∂  ∂  ∂  (A3) S U U S S Q the Ruppeiner invariant. 123 Eur. Phys. J. C (2017) 77:423 Page 19 of 21 423

References 26. P.C.W. Davies, Thermodynamic phase transitions of Kerr- Newman black holes in de Sitter space. Class. Quantum Grav. 1. J.D. Bekenstein, Black holes and the second law. Lett. Nuovo 6, 1909 (1989) Cim. 4, 737 (1972) 27. M. Sokolowski, P.Mazur, Second-order phase transitions in black- 2. J.D. Beckenstein, Black holes and entropy. Phys. Rev. D 7, 2333 hole thermodynamics. J. Phys. A 13, 1113 (1980) (1973) 28. L. Landau, E. M. Lifshitz, Statistical Physics. Pergamon Press, 3. J.M. Bardeen, B. Carter, S. Hawking, The four laws of black hole London, England (1980) mechanics. Commun. Math. Phys. 31, 161 (1973) 29. L. Landau, E. M. Lifshitz, Fluid Mechanics. Pergamon Press, 4. S.W. Hawking, Particle creation by black holes. Commun. Math. London, England (1959) Phys. 43, 199 (1975) 30. D. Pavon, J.M. Rubi, Nonequilibrium thermodynamic fluctuations 5. A. Strominger, C. Vafa, Microscopic orogin of the of black holes. Phys. Rev. D 37, 2052 (1988) BekensteinÐHawking entropy. Phys. Lett. B 379, 99 (1996). 31. D. Pavon, Phase transition in Reissner-Nordströ m black holes. arXiv:hep-th/9601029 Phys. Rev. D 43, 2495 (1991) 6. K. Skenderis, Black holes and branes in string theory. Lect. Notes 32. R.G. Cai, R.K. Su, P.K.N. Yu, Nonequilibrium thermodynamic Phys. 541, 325 (2000). arXiv:hep-th/9901050 fluctuations of charged dilaton black holes. Phys. Rev. D 48, 3473 7. S.D. Mathur, The fuzzball proposal for black holes: an elementary (1993) review. Fortsch. Phys. 53, 793 (2005). arXiv:hep-th/0502050 33. B. Wang, J.M. Zhu, Nonequilibrium thermodynamic fluctuations ( + ) 8. V.P.Frolov, D. V.Fursaev, Mechanism of generation of black hole of 2 1 -dimensional black holes. Mod. Phys. Lett. A 10, 1269 entropy in Sakharov’s Induced Gravity, Phys. Rev. D 56, 2212 (1995) (1997). arXiv:hepth/9703178 34. J. Shen, B. Wang,C.Y.Lin, R.G. Cai, R.K. Su, The phase transition 9. A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Quantum geom- and the Quasi-Normal Modes of black Holes. JHEP 0707, 037 etry and black hole entropy, Phys. Rev. Lett. 80, 904 (1998). (2007). arXiv:hep-th/0703102 arXiv:gr-qc/9710007 35. X. Rao, B. Wang, G. Yang, Quasinormal modes and phase 10. E. R. Livine, D. R. Terno, Quantum black holes: entropy and transition of black holes. Phys. Lett. B 649, 472 (2007). entanglement on the horizon, Nucl. Phys. B 741, 131 (2006). arXiv:0712.0645 arXiv:gr-qc/0508085 36. Y. Liu, D.C. Zou, B. Wang, Signature of the Van der Waals like 11. S. Carlip, Symmetries, horizons, and black hole entropy, Gen. small-large charged AdS black hole phase transition in quasinor- Rel .Grav. 39, 1519 (2007) (Int. J. Mod. Phys. D 17, 659 (2008)). mal modes. JHEP 1409, 179 (2014). arXiv:1405.2644 arXiv:0705.3024 37. X.-X. Zeng, H. Zhang, L.-F. Li, Phase transition of holographic 12. S. W. Wei, Y. X. Liu, Insight into the microscopic structure of entanglement entropy in massive gravity. Phys. Lett. B 756, 170 an AdS black hole from thermodynamical phase transition, Phys. (2016). arXiv:1511.00383 Rev. Lett. 115, 111302 (2015). arXiv:1502.00386 38. X.-X. Zeng, L.-F. Li, Van der Waals phase transition in the 13. G. Ruppeiner, Thermodynamics: a Riemannian geometric model. framework of holography. Phys. Lett. B 764, 100 (2017). Phys. Rev. A 20, 1608 (1979) arXiv:1512.08855 14. G. Ruppeiner, Riemannian geometry in thermodynamic fluctua- 39. X.-X. Zeng, X.-M. Liu, L.-F. Li, Phase structure of the Born- tion theory, Rev. Mod. Phys. 67, 605 (1995) [Erratum: Rev. Mod. Infeld-anti-de Sitter black holes probed by non-local observables. Phys. 68, 313 (1996)] Eur. Phys. J. C 76, 616 (2016). arXiv:1601.01160 15. H. Oshima, T. Obata, H. Hara, Riemann scalar curvature of ideal 40. J.-X. Mo, G.-Q. Li, Z.-T. Lin, X.-X. Zeng, Revisiting van der quantum gases obeying Gentile’s statistics. J. Phys. A: Math. Gen. Waals like behavior of f(R) AdS black holes via the two point cor- 32, 6373 (1999) relation function. Nucl. Phys. B 918, 11 (2017). arXiv:1604.08332 16. G. Ruppeiner, Thermodynamic curvature measures interactions. 41. S. He, L.-F. Li, X.-X. Zeng, Holographic Vander Waals-like phase Am. J. Phys. 78, 1170 (2010). arXiv:1007.2160 transition in the Gauss-Bonnet gravity. Nucl. Phys. B 915, 243 17. G. Ruppeiner, Thermodynamic curvature from the critical point to (2017). arXiv:1608.04208 the triple point. Phys. Rev. E 86, 021130 (2012). arXiv:1208.3265 42. X.-X. Zeng, L.-F. Li, Holographic phase transition probed by non- 18. H.O. May, P.Mausbach, G. Ruppeiner, Thermodynamic curvature local observables. Adv. High Energy Phys. 2016, 6153435 (2016). for attractive and repulsive intermolecular forces. Phys. Rev. E 88, arXiv:1609.06535 032123 (2013) 43. G. Q. Li, J. X. Mo, Phase transition and thermodynamic geometry 19. G. Ruppeiner, Thermodynamic curvature and phase transitions of f(R) AdS black holes in the grand canonical ensemble, Phys. in Kerr-Newman black holes. Phys. Rev. D 78, 024016 (2008). Rev. D 93, 124021 (2016). arXiv:1605.09121 arXiv:0802.1326 44. S. A. Hosseini Mansoori, B. Mirza, E. Sharifian, Extrinsic and 20. G. Ruppeiner, Thermodynamic curvature: pure fluids to black intrinsic curvatures in thermodynamic geometry, Phys. Lett. B holes. J. Phys. Conf. Series 410, 012138 (2013). arXiv:1210.2011 759, 298 (2016). arXiv:1602.03066 21. G. Ruppeiner, Thermodynamic curvature and black holes. 45. M. Chabab, H. El Moumni, K. Masmar, On thermodynamics of Springer Proc. Phys. 153, 179 (2014). arXiv:1309.0901 charged AdS black holes in extended phases space via M2-branes 22. M. Kord Zangeneh, A. Dehyadegari, A. Sheykhi, Comment on background, Eur. Phys. J. C 76, 304 (2016). arXiv:1512.07832 “Insight into the Microscopic Structure of an AdS Black Hole 46. S. Basak, P. Chaturvedi, P. Nandi, G. Sengupta, Thermodynamic from a Thermodynamical Phase Transition”. arXiv:1602.03711 geometry of holographic superconductors, Phys. Lett. B753, 493 23. A. Dehyadegari, A. Sheykhi, A. Montakhab, Microscopic Prop- (2016). arXiv:1509.00826 erties Of Black Holes Via An Alternative Extended Phase Space. 47. B. P. Dolan, The intrinsic curvature of thermodynamic poten- arXiv:1607.05333 tials for black holes with critical points, Phys. Rev. D 92, 044013 24. P.C.W. Davies, The Thermodynamic theory of black holes. Proc. (2015). arXiv:1504.02951 R. Soc. Lond. A 353, 499 (1977) 48. J. X. Mo, W. B. Liu, Non-extended phase space thermodynamics 25. P.C.W. Davies, Thermodynamics of black holes. Rep. Prog. Phys. of Lovelock AdS black holes in grand canonical ensemble, Eur. 41, 1313 (1978) Phys. J. C 75, 211 (2015). arXiv:1503.01956

123 423 Page 20 of 21 Eur. Phys. J. C (2017) 77:423

49. J. L. Zhang, R. G. Cai, H. Yu, Phase transition and thermodynami- 72. J. M. Maldacena, The large-N limit of superconformal field the- cal geometry of Reissner-Nordström-AdS black holes in extended ories and super-gravity, Adv. Theor. Math. Phys. 2, 231 (1998) phase space, Phys. Rev. D 91, 044028 (2015). arXiv:1502.01428 (Int. J. Theor. Phys. 38, 1113 (1999)). arXiv:hep-th/9711200 50. S.A. Hosseini Mansoori, B. Mirza, M. Fazel, Hessian matrix, spe- 73. E. Witten, Anti de Sitter space and holography. Adv. Theor. Math. cific heats, Nambu brackets, and thermodynamic geometry. JHEP Phys. 2, 253 (1998). arXiv:hep-th/9802150 1504, 115 (2015). arXiv:1411.2582 74. S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory cor- 51. J.L. Zhang, R.G. Cai, H. Yu, Phase transition and thermodynam- relators from non-critical string theory. Phys. Lett. B 428, 105 5 ical geometry for Schwarzschild AdS black hole in AdS5 × S (1998). arXiv:hep-th/9802109 spacetime. JHEP 1502, 143 (2015). arXiv:1409.5305 75. M. Kord Zangeneh, A. Dehyadegari, A. Sheykhi, M. H. Dehghani, 52. R. Tharanath, J. Suresh, N. Varghese, V.C. Kuriakose, Ther- Thermodynamics and gauge/gravity duality for Lifshitz black modynamic Geometry of Reissener-Nordström-de Sitter black holes in the presence of exponential electrodynamics, JHEP 1603, hole and its extremal case. Gen. Relat. Grav. 46, 1743 (2014). 037 (2016). arXiv:1601.04732 arXiv:1404.6789 76. A. Dehyadegaria, A. Sheykhi and M. Kord Zangeneh, Holo- 53. J. Suresh, R. Tharanath, N. Varghese, V.C. Kuriakose, Thermo- graphic conductivity for logarithmic charged dilaton-Lifshitz dynamics and thermodynamic geometry of Park black hole. Eur. solutions, Phys. Lett. B 758, 226 (2016). arXiv:1602.08476 Phys. J. C 74, 2819 (2014). arXiv:1403.4710 77. E.S. Fradkin, A.A. Tseytlin, Effective field theory from quantized 54. S.A. Hosseini Mansoori, B. Mirza, Correspondence of phase tran- string. Phys. Lett. B 163, 123 (1985) sition points and singularities of thermodynamic geometry of 78. R.R. Metsaev, M.A. Rakhmanov, A.A. Tseytlin, The BornÐInfeld black holes. Eur. Phys. J. C 74, 2681 (2014). arXiv:1308.1543 action as the effective action in the open superstring theory. Phys. 55. S. Kachru, X. Liu, M. Mulligan, Gravity Duals of Lifshitz-like Lett. B 193, 207 (1987) Fixed Points. Phys. Rev. D 78, 106005 (2008). arXiv:0808.1725 79. E. Bergshoeff, E. Sezgin, C. Pope, P. Townsend, The BornÐlnfeld 56. G. Bertoldi, B.A. Burrington, A. Peet, Black holes in asymptot- action from conformal invariance of the open superstring. Phys. ically Lifshitz spacetimes with arbitrary critical exponent. Phys. Lett. B 188, 70 (1987) Rev. D 80, 126003 (2009). arXiv:0905.3183 80. C.G. Callan, C. Lovelace, C.R. Nappi, S.A. Yost,Loop corrections 57. M.H. Dehghani, R.B. Mann, LovelockÐLifshitz Black Holes. to superstring equations of motion. Nucl. Phys. B 308, 221 (1988) JHEP 1007, 019 (2010). arXiv:1004.4397 81. O.D. Andreev, A.A. Tseytlin, Partition-function representation for 58. M.H. Dehghani, R.B. Mann, Thermodynamics of LovelockÐ the open superstring effective action: cancellation of M öbius Lifshitz black branes. Phys. Rev. D 82, 064019 (2010). infinites and derivative corrections to Born-lnfeld lagrangian. arXiv:1006.3510 Nucl. Phys. B 311, 205 (1988) 59. M.H. Dehghani, Sh Asnafi, Thermodynamics of rotating 82. R.G. Leigh, Dirac-BornÐlnfeld action from Dirichlet σ-model. LovelockÐLifshitz black branes. Phys. Rev. D 84, 064038 (2011). Mod. Phys. Lett. A 4, 2767 (1989) arXiv:1107.3354 83. M. Born, L. lnfeld, Foundation of the new field theory, Proc. R. 60. M.H. Dehghani, Ch. Shakuri, M.H. Vahidinia, Lifshitz black Soc. A 144, 425 (1934) brane thermodynamics in the presence of a nonlinear electromag- 84. S.W. Hawking, D.N. Page, Thermodynamics of black holes in netic field. Phys. Rev. D 87, 084013 (2013). arXiv:1306.4501 anti-De Sitter Space. Commun. Math. Phys. 87, 577 (1983) 61. M. Bravo-Gaete, M. Hassaine, Thermodynamics of charged Lif- 85. I. Amado, A.F. Faedo, Lifshitz black holes in string theory. JHEP shitz black holes with quadratic corrections, Phys. Rev. D 91, 1107, 004 (2011). arXiv:1105.4862 064038 (2015). arXiv:1501.03348 86. Y.S. Myung, Phase transitions for the Lifshitz black holes. Eur. 62. A. Alvarez, E. Ayon-Beato, H.A. Gonzalez, M. Hassaine, Non- Phys. J. C 72, 2116 (2012). arXiv:1203.1367 linearly charged Lifshitz black holes for any exponent z>1. JHEP 87. M. Ghodrati, A. Naseh, Phase Transitions in BHT Massive Grav- 1406, 041 (2014). arXiv:1403.5985 ity. arXiv:1601.04403 63. M. Taylor, Non-Relativistic Holography. arXiv:0812.0530 88. R.B. Mann, Lifshitz topological black holes. JHEP 0906, 075 64. J. Tarrio, S. Vandoren, Black holes and black branes in Lifshitz (2009). arXiv:0905.1136 spacetimes. JHEP 1109, 017 (2011). arXiv:1105.6335 89. A. Sheykhi, Thermodynamical properties of topological BornÐ 65. J. Polchinski, String Theory (Cambridge University Press, Cam- Infeld-dilaton black holes. Int. J. Mod. Phys. D 18, 25 (2009). bridge, 1998) arXiv:0801.4112 66. G. Bertoldi, B.A. Burrington, A.W. Peet, Thermal behavior of 90. A. Sheykhi, Thermodynamics of charged topological dilaton charged dilatonic black branes in AdS and UV completions black holes. Phys. Rev. D 76, 124025 (2007). arXiv:0709.3619 of Lifshitz-like geometries. Phys. Rev. D 82, 106013 (2010). 91. S.W. Hawking, Black hole explosions. Nature (London) 248,30 arXiv:1007.1464 (1974) 67. M. Kord Zangeneh, M. H. Dehghani, A. Sheykhi, Thermodynam- 92. G.W.Gibbons, S.W.Hawking, Action integrals and partition func- ics of Gauss-Bonnet-Dilaton Lifshitz Black Branes, Phys. Rev. D tions in quantum gravity. Phys. Rev. D 15, 2738 (1977) 92, 064023 (2015). arXiv:1506.07068 93. C. J. Hunter, Action of instantons with a nut charge, Phys. Rev. D 68. M. Hassaine, C. Martinez, Higher-dimensional black holes with a 59 , 024009 (1999). arXiv:1506.01784 conformally invariant Maxwell source. Phys. Rev. D 75, 027502 94. S.W. Hawking, C.J. Hunter, D.N. Page, N.U.T. Charge, anti- (2007). arXiv:hep-th/0701058 de Sitter space, and entropy. Phys. Rev. D 59, 044033 (1999). 69. H.H. Soleng, Charged blach point in general relativity coupled to arXiv:hep-th/9809035 the logarithmic U(1) gauge theory. Phys. Rev. D 52, 6178 (1995). 95. R.B. Mann, Misner string entropy. Phys. Rev. D 60, 104047 arXiv:hep-th/9509033 (1999). arXiv:hep-th/9903229 70. S.H. Hendi, Asymptotic charged BTZ black hole solutions. JHEP 96. R.B. Mann, Entropy of rotating Misner string spacetimes. Phys. 1203, 065 (2012). arXiv:1405.4941 Rev. D 61, 084013 (2000). arXiv:hep-th/9904148 71. M. Kord Zangeneh, A. Sheykhi, M. H. Dehghani, Thermodynam- 97. J. Brown, J. York, Quasilocal energy and conserved charges ics of topological nonlinear charged Lifshitz black holes, Phys. derived from the gravitational action, Phys. Rev. D 47, 1407 Rev. D 92, 024050 (2015). arXiv:1506.01784 (1993). arXiv:gr-qc/9209012

123 Eur. Phys. J. C (2017) 77:423 Page 21 of 21 423

98. J.D. Brown, J. Creighton, R. B. Mann, Temperature, energy, and 102. S. H. Hendi, S. Panahiyan, B. Eslam Panah, Geometrical method heat capacity of asymptotically anti-de Sitter black holes, Phys. for thermal instability of nonlinearly charged BTZ Black Holes, Rev. D 50, 6394 (1994). arXiv:gr-qc/9405007 Adv. High Energy Phys. 2015, 743086 (2015). arXiv:1509.07014 99. S.H. Hendi, A. Sheykhi, M.H. Dehghani, Thermodynamics of 103. W.A. Hiscock, L.D. Weems, Evolution of charged evaporating higher dimensional topological charged AdS black branes in dila- black holes. Phys. Rev. D 41, 1142 (1990) ton gravity. Eur. Phys. J. C 70, 703 (2010). arXiv:1002.0202 104. J.S. Schwinger, On gauge invariance and vacuum polarization. 100. H. B. Callen, Thermodynamics and an Introduction to Thermo- Phys. Rev. 82, 664 (1951) statics, Wiley, New York (1985) 101. M. Kardar, Statistical Physics of Particles (Cambridge University Press, Cambridge, 2007)

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