Thermodynamic curvature of the Schwarzschild-AdS black hole and Bose condensation
CChemical potential, microstructures and phase transition of the Schwarzschild-AdS black hole
Sandip Mahish ∗ , Aritra Ghosh † and Chandrasekhar Bhamidipati ‡ School of Basic Sciences, Indian Institute of Technology Bhubaneswar,Jatni, Khurda, Odisha, 752050, India
Abstract
In the AdS/CFT correspondence, a variable cosmological constant Λ in the bulk cor-responds to varying the number of colors N in the boundary gauge theory, with chemicalpotential µ as its thermodynamic conjugate. In this work, within the context of AdS × S and its dual N = 4 SUSY Yang-Mills theory at large N , we investigate the nature of mi-crostructures and phase transitions by studying the Ruppeiner curvature R . We find thatthe large black hole branch is associated with purely attractive microstructures and itsbehaviour bears some qualitative similarities with that of an ideal Bose gas ( R < µ < R takes increasingly negative values signifying long range correlations and strongquantum fluctuations. The small black hole branch on the other hand, is attractive at lowtemperatures and repulsive at high temperatures with a second order critical point whichroughly separates the two regions. The scaling behaviour associated with the specific heatand the thermodynamic curvature around this second order critical point is obtained. The formulation of the laws of black hole mechanics in direct analogy with those of thermody-namics [1]-[6] has paved the way for a large body of work in this subject in last few decades. Inparticular, the Hawking-Page (HP) transition [7] in black holes asymptotic to AdS spacetimeshave attracted much attention with interpretation involving AdS/CFT correspondence [8]-[11],as a confinement-deconfinement phase transition from the dual field theory side [11]. Particu-larly, the study of black hole phase transitions continues to provide a rich arena for exploringthe connections between gravity and gauge theories (see [12, 13]). ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] a r X i v : . [ h e p - t h ] J un he proposal of treating the negative cosmological constant of the AdS spacetime asdynamical (starting from earlier works in literature, e.g., [14]-[21]) has lead to a flurry ofresearch activity over the last decade. However, the interpretation of the cosmological constantΛ as a pressure P and subsequently associating the notion of a thermodynamic volume with ablack hole horizon has been a rather recent development [22] (see also [23, 24]), with at leasttwo possible approaches. In the first approach, quite remarkably, for non-rotating black holes infour or higher dimensions, this thermodynamic volume V exactly corresponds to the geometricvolume of the black hole horizon. In this framework, most black holes have been associatedwith a P − V critical behaviour similar to that of the van der Waals’ fluid (see for example[25]-[28]) with enthalpy H ( S, P ) playing the central role, where, S is entropy. However, a dualfield theoretic interpretation of such phase transitions is not developed yet and one could ask,what is the interpretation of the dynamical cosmological constant on the dual conformal fieldtheory (CFT) at the boundary? To this end, there is a second approach which gives a morenatural interpretation of a variable cosmological constant as arising from a variation of numberof colours of the boundary CFT or equivalently, related to the number N of branes within theAdS bulk [8] (also see [29, 30]). Progress in this approach has been slow, though the settingis very interesting as one can understand the physics of CFT a bit better. In this context,it is possible to envisage a situation where one introduces a chemical potential µ conjugateto a ‘variable’ N , to go with other thermodynamical potentials (such as, electric potential forcharged systems and so on). The study of the chemical potential µ in this set up is expectedto be interesting. For instance, if one invokes the microscopic interpretation from standardthermodynamics, µ is known to be negative for a classical ideal Bose gas and positive if thereis a sufficiently strong repulsive interaction among particles or in fermionic systems at lowtemperatures. In particular, µ can be zero when the average thermal de Broglie wavelength ofparticles is comparable to the inter particle separation, signifying the onset of quantum effectsin the system. Indeed, in a remarkable suggestion in the context of Schwarzschild black holesin AdS [30], Dolan noted that the temperature at which µ approaches zero is close to the pointwhere the system undergoes Hawking-Page transition and proposed this to correspond to Bosecondensation. This was improved further in [33], who showed that µ = 0 might be reached invarious black holes in AdS, above the HP transition temperature, if one makes use of densitiesinstead of absolute quantities.On the otherhand, the use of Ruppeiner geometry in studying thermodynamics and phasetransitions is well known, with the metric defined as [36], g ij = − ∂ i ∂ j S, (1.1)where S is the entropy of the system. At a finite temperature, the associated curvature scalartypically diverges when the system approaches a phase transition point. Ruppeiner geometryhas been worked out for a wide range of thermodynamic systems including magnetic systems[37, 38] and quantum gases [39]. The magnitude of Ruppeiner curvature has been shown toprovide physical insights on fluctuations and the stability of a given system [39]-[41]. Divergenceof the Ruppeiner curvature indicates a strongly correlated behaviour among the microstructures(see for example [41]). In black hole thermodynamics, Ruppeiner geometry was first studied inref [42] over two decades ago where it was applied to understand the thermodynamic behaviour2f BTZ black holes. Subsequently, the Ruppeiner geometry for RN, Kerr and RN-AdS blackholes has been studied in several works (see for example [43]-[46]; also see [47]) and it was alsoshown that the divergence of the curvature scalar is consistent with the Davies phase transitionpoint [48] (also see [49, 50]). However, more recently there has been a considerable interestin the use of Ruppeiner geometry for probing the nature of microscopic interactions (see forexample [51]-[67] for some selected recent works and references therein) in black holes, mostlyfrom the dynamical cosmological constant picture. Since it is well known that black holes havea temperature and an entropy, it is suggestive to associate microstructures with them whichshare the degrees of freedom of the entropy. A positive Ruppeiner curvature has been shownto indicate dominance of repulsive interactions as is the case of charged and/or rotating BTZblack holes (see [64, 65] and references therein) whereas, a negative one indicates attractiveinteractions as in the case of Schwarzschild-AdS black holes [58]. Black holes in four or moredimensions carrying electric charge have been shown to be associated with both attractive andrepulsive interactions among the microstructures [59]-[63]. However for AdS black holes, mostof this work has been done in the setup where the cosmological constant is interpreted as thethermodynamic pressure. The primary motivation of this work is to fill the gap by studyingthe behaviour of the black hole microstructures in the alternate second approach, when thecosmological constant is related to the number of colours on the boundary CFT and we find aremarkable behaviour associated with both the small and large Schwarzschild-AdS black hole.In this work, we set up Ruppeiner geometry in the novel ( s, N )-plane, where s is theentropy density, to probe the behaviour of the interacting black hole microstructures and alsophase transitions in Schwarzschild-AdS black holes. Let us note that methods of Ruppeinergeometry have been applied earlier to studying phase transitions and critical behaviour ofSchwarzschild and Reissner-Nordstr¨om black holes [31, 32, 34]. In particular, in [31], differ-ent metrics such as the ones given by Ruppeiner, Weinhold and Quevedo were used to checkwhether thermodynamic geometry captures the zeros and divergences of heat capacity at con-stant chemical potential. However, as mentioned above, the issue of microstructures and thenature of their interactions, along the lines of recent works on the ( T, V )- and (
S, P )- planes[57]-[67] has not been undertaken yet in the new plane being studied here, where the pairs(
S, N ) or ( T, µ ) are chosen as fluctuation coordinates. As we show in this work, the study ofthermodynamic geometry in these new planes is quite important as apart from capturing thedivergences of specific heats, the classification of attraction and repulsion dominated regions isuseful while probing the nature of phase transitions.The plan of the rest of the paper is as follows. In the following section-(2), we setup the thermodynamics for the five dimensional Schwarzschild-AdS and present some resultswhich would be required for our analysis of Ruppeiner geometry. In particular, we carefullyscrutinise the behaviour of the chemical potential and the associated fugacity function. Section-(3), contains the main results of this work, with the expression and plots of the Ruppeinercurvature and its physical interpretation primarily in the context of black hole microstructures.It is shown that the large black hole branch qualitatively resembles an ideal Bose gas withfluctuations getting significant as the fugacity approaches unity. We also analyze the secondorder critical point in the small black hole branch. Finally, the work is concluded with someremarks in section-(4). 3
Thermodynamics with chemical potential
We shall for our purposes of this paper, consider the Schwarzschild black hole in
AdS comingfrom the D = 10 maximal supergravity which is compactified on S . More explicitly, oneconsiders the type IIB supergravity on AdS × S which corresponds to the thermal N = 4supersymmetric Yang-Mills (SYM) theory at large N on the dual side. The decoupled tendimensional metric takes the form (see for example [12]), ds = ds + l (cid:88) i =1 (cid:2) dθ i + θ i ( dφ i + 2 √ A ν dx ν ) (cid:3) , (2.1)where ν = 0 , , , ,
4. The terms within the summation correspond to the S part with { θ i } being the direction cosines and { φ i } being the rotation angles on S whereas ds is theSchwarzschild metric asymptotic to AdS given as, ds = − f ( r ) dt + f ( r ) − dr + r d Ω , (2.2)with d Ω being the line element on the maximally symmetric Einstein space in three dimensionswith a spherical topology and enclosing a unit volume. The lapse function f ( r ) is, f ( r ) = 1 − G (5) M πr + r l , (2.3)where l is the radius of the AdS spacetime and G (5) is Newton’s constant in five dimensions.The black hole mass is calculated from the equation, f ( r + ) = 0 where r + is the event horizonradius. Subsequently, the thermodynamics of the five dimensional Schwarzschild-AdS blackhole can be discussed setting the mass to be equal to an appropriate thermodynamic potential,as is done in black hole thermodynamics.We now briefly discuss some thermodynamic features of the five dimensional Schwarzschild-AdSblack hole . The black hole mass is obtained from the metric to be, M = 3 π r l G (10) (cid:18) r l (cid:19) , (2.4)where G (10) is the fixed Newton’s constant in ten dimensions which we from now on set equalto unity, i.e. G (10) = 1. The entropy of the black hole is given by the Bekenstein-Hawkingentropy formula which in d = 5 gives , S = π r G (5) = π l r . (2.5) Being direction cosines, they are not all independent and satisfy (cid:80) i =1 θ i = 1. Some of these discussions have appeared earlier in the literature. See [30]-[34] for some earlier works. The Newton’s constant in five dimensions is related to that in ten dimensions as, G (5) = G (10) /π l .Therefore, G (5) depends on l and is not strictly constant. We shall set (cid:126) = k B = 1 for all subsequent discussions. Consequently, the (fixed) ten dimensional Plancklength, l P = (cid:126) G (10) = 1 and G (5) = G (10) /π l = 1 /π l . N is related to l as, N = π l √ . (2.6)This means that a dynamical Λ leads to the notion of varying the number of branes in thebulk. On the boundary CFT however, which in this case is the N = 4 SYM theory, N is therank of the gauge group SU ( N ). Noting that the number of degrees of freedom in the large N limit scales as N rather than N [35], we would consider N as the thermodynamic variableassociated with the dynamical cosmological constant.The black hole mass in eqn (2.4) can be written down as a function of S and N using eqns(2.5) and (2.6) as: M = M ( S, N ). The first law of black hole thermodynamics reads, dM = T dS + µdN , (2.7)where µ is a suitable thermodynamic conjugate to N and bears its interpretation as a chemicalpotential. Noting however that N corresponds to the number of degrees of freedom which areassociated with an effective energy µN , we define the internal energy of the black hole as U = M − µN . This means that one can write the black hole mass as, M = U + µN , whichbears a close resemblance with the identification of the black hole mass with its enthalpy as isdone in conventional extended black hole thermodynamics [22]. We would, following [32, 33]consider for our purposes, the densities of the extensive thermodynamic quantities M (andhence all other thermodynamic potentials defined via Legendre transformations of M ) and S by scaling them down by a factor of l since for the five dimensional bulk of AdS its dualCFT is associated with a volume which apart from constant factors goes as, V ∼ l . The massdensity ρ is then given as, ρ = 3 s / (cid:0) N / + 2 √ s / (cid:1) / πN / , (2.8)which satisfies the first law of black hole thermodynamics expressed in terms of densities , dρ = T ds + µdN , (2.9)where s is the entropy density obtained from eqn (2.5). The Hawking temperature is given by, T = (cid:18) ∂ρ∂s (cid:19) N = N / + 4 √ s / × / πN / √ s . (2.10)The Hawking temperature has a minimum value, T min which occurs at s = 0 . T min , there are two values of s with the same temperature: s < . s > . Note that the µ appearing in eqns (2.7) and (2.9) cannot have the same functional form because we haven’tscaled down N by a factor of l . They are naively labelled the same just for notational ease and for the restof the paper, we would be considering the chemical potential µ to be the one defined from eqn (2.9). .1 Chemical potential Since, a density of N has not been considered, then for the first law [eqn(2.9)] to hold goodone has to define the chemical potential as, µ =: (cid:18) ∂ρ∂N (cid:19) s = √ N / s / − × / s / πN / . (2.11)Figure-(1) shows the variation of the chemical potential as a function of the temperature. Thelarge and small black hole branches are indicated. The chemical potential is negative definite - - - - - μ μ vs T: AdS Schwarzschild ( D = ) Small Black hole BranchLarge Black hole Branch
Figure 1: Chemical potential of the five dimensional Schwarzschild-AdS black hole as a functionof the Hawking temperature with fixed N .in the large black hole branch and monotonically goes to negative infinity as T → ∞ (orequivalently s → ∞ ). It can be checked that the chemical potential goes to zero exactly atthe Hawking-Page temperature , T HP = 0 . s HP = 0 .
25 whereas, the point µ = 0 lies in the small black hole branchwith s µ =0 = 0 . −∞ > µ > z = e βµ , (2.12)where β = 1 /T is the inverse temperature factor. For a quantum gas, the limit z → T → ∞ or equivalently µ → −∞ . The other extreme limit foran ideal gas of Bosons is z → For general N , one has T HP = 0 . N − / . We will set N = 1 in subsequent plots. z ∈ (0 , Fugacity ( z ) vs T: AdS Schwarzschild ( D = ) Large Black Hole Branch
Figure 2: Fugacity of the five dimensional Schwarzschild-AdS large black hole as a function ofthe Hawking temperature. µ = 0 does not lie in the large black hole branch, the fugacity does not exactly reach unity. Con-sequently, there is no Bose condensation-like phenomena for the large black hole. Rather, thelimit z → s µ =0 = 0 . s < s µ =0 , the chemical potential is positive and vice versa. In terms of the Hawkingtemperature, black holes colder than the µ = 0 case are associated with negative chemicalpotential whereas the hotter ones have positive chemical potentials. The fugacity is plottedin figure-(3) and reaches unity exactly at the point where µ = 0 which corresponds to thetemperature T µ =0 = 0 . T > T µ =0 , the fugacity is positive and close to unity while if T min < T < T µ =0 , fugacity can get negative (not shown in the plot) but is still close to unity. We shall next briefly comment on the behaviour of the specific heats. From the first law [eqn(2.9)], one can obtain the specific heat density at constant N as, C N = (cid:18) ∂ρ∂T (cid:19) N = − s (cid:0) N / + 4 √ s / (cid:1) N / − √ s / . (2.13) Incidentally, this temperature happens to have the same numerical value as T HP . Since we are working with densities of the thermodynamic potentials, the specific heat so defined is essentiallythe specific heat density. Having said that, we will from now on refer to them simply as specific heats expectingthe reader to understand without confusion that a density it is implied. .5 1.0 1.5 2.0T1.00001.00011.00021.00031.00041.00051.0006z Fugacity ( z ) vs T: AdS Schwarzschild ( D = ) Small Black Hole Branch
Figure 3: Fugacity of the five dimensional Schwarzschild-AdS large small hole as a function ofthe Hawking temperature.which is plotted as a function of the temperature in figure-(4). It has a divergence at T = T min and is positive for large black hole while it is negative for the small black hole as also noted in [7].The point of divergence therefore separates the positive and negative branches of the specificheat C N . Moreover, since from eqn (2.9), one can also define C N as the second derivative ofthe Gibbs free energy density g = ρ − T s as, C N = − T (cid:18) ∂ g∂T (cid:19) N , (2.14)one concludes that the divergence of C N is reminiscent of a second order phase transition,much like that pointed out by Davies [48] which separates the positive and negative regions ofthe specific heat. Particularly of interest in our case is the specific heat at constant chemical - - C N C N vs T: AdS Schwarzschild ( D = ) Small Black Hole BranchLarge Black Hole Branch
Figure 4: Specific heat at constant N for the five dimensional Schwarzschild-AdS black holeas a function of the Hawking temperature.potential, i.e. C µ . We define the internal energy density of the black hole from its mass density8s, u = ρ − µN and then the first law [eqn (2.9)] is modified to, du = T ds − N dµ, (2.15)leading to the straightforward definition of C µ as, C µ = (cid:18) ∂u∂T (cid:19) µ , (2.16)or equivalently, C µ = − T (cid:18) ∂ f∂T (cid:19) µ , (2.17)where f = u − T s is the Helmholtz free energy density. Subsequently, C µ can be calculated togive, C µ = − −
512 2 / s / N / + 11 N / s − √ s / N / − √ s / . (2.18)The variation of C µ as a function of temperature is shown in figure-(5). We should bear in - - C μ C μ vs T: AdS Schwarzschild ( D = ) Small Black Hole BranchLarge Black Hole Branch
Figure 5: Specific heat at constant µ for the five dimensional Schwarzschild-AdS black hole asa function of the Hawking temperature.mind that C µ is not the standard specific heat of the black hole defined directly from the mass(in this case, the mass density) and therefore, its behaviour is expected to be atypical. As canbe seen from figure-(5), it is almost always negative for both the branches. Further, exists apoint of divergence at T C µ =0 = 0 . Ruppeiner geometry, interacting microstructures andphase transitions
We shall now study the Ruppeiner geometry for the five dimensional Schwarzschild-AdS blackhole whose thermodynamics was briefly explored in the previous section. We note that theRuppeiner metric is defined as the negative Hessian of the entropy (here, entropy density) sothat the corresponding line element becomes, dl R = − ∂ s∂x i ∂x j dx i dx j , (3.1)where i, j ∈ { , , ...., n } and { x i } are independent coordinates in the thermodynamic phasespace or more precisely the subspace of thermodynamic equilibrium states which are allowedto fluctuate. For our purposes, we shall be require n = 2, i.e. two fluctuation coordinates inthe thermodynamic phase space. In that case, the metric shall have components denoted by g , g , g and g with g = g as a consequence of the symmetry of the metric tensor. Theline element can then be expressed as, dl R = g ( dx ) + 2 g ( dx )( dx ) + g ( dx ) , (3.2)with, g ij = − ∂ s∂x i ∂x j , i, j = 1 , . (3.3)For such a two dimensional metric, the scalar curvature can be written down as [68], R = − √ g (cid:20) ∂∂x (cid:18) g g √ g ∂g ∂x − √ g ∂g ∂x (cid:19) + ∂∂x (cid:18) √ g ∂g ∂x − √ g ∂g ∂x − g g √ g ∂g ∂x (cid:19)(cid:21) . (3.4)Here g is the determinant of the metric tensor. A straightforward comparison of the first lawof conventional extended black hole thermodynamics, i.e. dM = T dS + V dP with eqn (2.7)leads to the correspondence ( µ, N ) → ( V, P ). We shall perform our calculations on the directanalogue of the (
S, P )-plane, which in this case and in terms of densities is the ( s, N )-plane,i.e. with fluctuation coordinates s and N . The fundamental thermodynamic potential is ρ = ρ ( s, N ) and the line element can be expressed without much difficulty as, dl R = 1 C N ( ds ) + 2 T (cid:18) ∂T∂N (cid:19) s ( ds )( dN ) + 1 T (cid:18) ∂µ∂N (cid:19) s ( dN ) . (3.5)The analogue of the ( T, V )-plane in our case is the (
T, µ )-plane on which the thermodynamicpotential is f ( T, µ ) = u − T s = ρ − µN − T s . The line element is obtained to be, dl R = − C µ T ( dT ) − T (cid:18) ∂N ∂T (cid:19) µ ( dT )( dµ ) − T (cid:18) ∂N ∂µ (cid:19) T ( dµ ) . (3.6) More geometrically, the thermodynamic phase space is five dimensional in this case and spaces the thermo-dynamic equilibrium states corresponding to the system are all two dimensional Legendre submanifolds (see forexample [63]). s, µ ) and (
T, N ) as fluctuation coor-dinates . The curvature scalars can be shown to be equivalent by direct calculation and hencewe use the notation R without explicit reference to the plane on which it is calculated. Thedivergence of the Ruppeiner curvature can be shown to signal critical behaviour (or extremalityif one considers charged black holes) if it exists for a black hole. In the limit that R → ∞ ,it is therefore expected that the system gets strongly correlated. Indeed, it is now known [41]that | R | ∼ ξ ˜ d for a system of ˜ d spatial dimensions with ξ being the correlation length meaningthat the divergence of the Ruppeiner curvature can therefore be used as a test probe find outpossible critical points in a black hole system.It has long been observed that R = 0 for non-interacting systems such as the ideal gas whereas,a non-zero Ruppeiner curvature indicates presence of non-trivial interactions. For systems withdominant attraction, such as a van der Waals gas or an ideal Bose gas, one gets R <
R >
0. This allows one to probe usingthe sign of R , the nature of dominant interactions among black hole microstructures. Further,since it is well known that the Ruppeiner metric is closely related to thermodynamic fluctua-tions , it has been argued (see for example [39] although their sign convention is opposite toours). that the Ruppeiner curvature R be used as a measure of the stability of a thermodynamicsystem against fluctuations. We shall next closely examine the physical implications of R forthe five dimensional Schwarzschild-AdS black hole. D = 5 Schwarzschild-AdS black hole
For the present case, the Ruppeiner curvature can be calculated to be, R = 8 (cid:0)
40 2 / N / s / + 160 N / s / − √ N / + 768 √ s (cid:1) N / √ s (cid:0) N / − √ s / (cid:1) (cid:0) N / + 4 √ s / (cid:1) . (3.9)It has been plotted as a function of s is figure-(6). The curvature initially starts out as posi-tive for small values of s within the small black hole branch and has an infinite discontinuityat the point s R = ∞ = 0 . C µ diverges.The Ruppeiner curvature then has a zero crossing at s R =0 = 0 . On the ( s, µ )-plane, the thermodynamic potential is u ( s, µ ) = ρ − µN and the line element reads, dl R = 1 C µ ( ds ) + 2 T (cid:18) ∂T∂µ (cid:19) s ( ds )( dµ ) − T (cid:18) ∂N ∂µ (cid:19) s ( dµ ) , (3.7)whereas, on the ( T, N )-plane the thermodynamic potential is, g ( T, N ) = ρ − T s and the corresponding lineelement can be shown to be given as, dl R = − C N T ( dT ) − T (cid:18) ∂s∂N (cid:19) T ( dT )( dN ) + 1 T (cid:18) ∂µ∂N (cid:19) T ( dN ) . (3.8) In fact, the notion of the Ruppeiner metric can be derived by inverting the well known relation, S = ln Ω.See [62] for instance. .01 0.02 0.03 0.04s50010001500R R vs s: AdS Schwarzschild ( D = ) N = Figure 6: Ruppeiner curvature for the five dimensional Schwarzschild-AdS black hole as afunction of the entropy density with fixed N .negative for higher values of s . All of this happens in the small black hole branch while thechemical potential is positive. For an even higher entropy density s µ =0 = 0 . s = 0 . s HP = 0 .
25 which corresponds tothe r + = l point. The Ruppeiner curvature can also be plotted with respect to the temperaturein which the small and large black hole branches can be seen separately, as is shown in figure-(7). Small Black Hole Branch - R vs T: AdS Schwarzschild ( D = ) Small Black Hole BranchLarge Black Hole Branch
Figure 7: Ruppeiner curvature for the five dimensional Schwarzschild-AdS black hole as afunction of Hawking temperature with fixed N .As can be easily seen, nothing very remarkable happens in the large black hole branch as faras Ruppeiner curvature is concerned even though the Hawking-Page transition happens in this12 .992 0.996 1 z - -
10R 0.2 0.4 0.6 0.8 1.0 z - - - - R vs fugacity ( z ) : AdS Schwarzschild ( D = ) Figure 8: Ruppeiner curvature for the five dimensional Schwarzschild-AdS large black hole asa function of fugacity with fixed N .branch . The Ruppeiner curvature is negative for the large black hole indicating presence ofattractive interactions among the black hole microstructures. Moreover, the chemical potentialis negative definite in this branch tempting us to interpret the Schwarzschild-AdS large blackhole as the black hole analogue of the ideal Bose gas. The curvature scalar asymptotically goesto zero as T → ∞ (equivalently s → ∞ ) which results in the fugacity, z → R = 0. In the low temperature limit of thelarge black hole, one has z → z approaches unity, R rapidly takes large negativevalues. This can be taken to indicate the presence of strong quantum fluctuations leading tolong range correlations within the system as was argued in ref [39]. However, the case µ = 0 isnot reached by the large black hole branch and therefore there is no Bose condensation involved.We shall next discuss the small black hole branch, which seems to be more interesting. As faras nature of interactions among the microstructures is concerned, the small black hole startsout as a repulsive system at very small s , i.e. high T . Exactly at s R = ∞ = 0 . C µ . Since the systemhas no extremal point, such a divergence is expected to signal phase transition of the blackhole . Such a point can be therefore be interpreted as a second order critical point. It shouldbe remarked that there are no first order lines which terminate at the second order critical The Ruppeiner curvature doesn’t seem to capture the physics of the Hawking-Page transition. A similar remark was made earlier in ref [31]. However, no further investigation on the nature of this pointwas pursued. R divergesis given by T c = 0 . C µ given in eqn (2.18) in terms of the temperature as, C µ == − √ A / N / −
168 2 / A / + 176 AN /
384 (3 2 / A / − N / ) , (3.10)where, A = − N / (cid:114)(cid:16) π √ N T − (cid:17) (cid:16) π √ N T − (cid:17) − πN / T + 2 π N T , (3.11)it can be verified that upon defining t = T /T c −
1, the specific heat diverges at t = 0 as, C µ ∼ | t | − , (3.12)which gives the value of the critical exponent α = 1. It is also straightforward to show that theRuppeiner curvature exhibits the following scaling behaviour around the critical point t = 0, R ∼ | t | − . (3.13)which is the same as that noted in refs [34, 60, 69, 70] in slightly different contexts. Conse-quently, we expect that the correlation length also scales with temperature around the criticalpoint. The Schwarzschild-AdS small black hole therefore, admits a second order critical pointaround which both the specific heat C µ and the Ruppeiner curvature R admit scaling behaviourand diverge exactly at the critical temperature. Further, at a slightly lower entropy density s R =0 = 0 . T R =0 = 0 . T c − T R =0 T c = 0 . , (3.14)i.e. approximately by 3 . T R =0 < T < T min , attrac-tive interactions are dominant. Therefore, the small black hole is repulsive at high temperatureswhile it is attractive at low temperatures with a switching between the two regions taking placeexactly at T = T R =0 . While all of this happens in the µ > s ), the relative numberdensities of microstructures of the two kinds would dictate which kind of interaction would bedominant in the system explaining the presence of both repulsion and attraction dominatedregions. The critical exponent α is defined as, C ∼ | T − T c | − α . Remarks
In this work, we have studied the thermodynamic geometry of the five dimensional Schwarzschild-AdS black hole in the context where the dynamical cosmological constant is treated to corre-spond to number of colours in the dual gauge theory. The main focus was to get an under-standing of the nature of interactions of microstructures through the behaviour of Ruppeinercurvature and comparing it with the behaviour of chemical potential, for large and small blackhole branches of the black hole, respectively. Our results are summarised as follows, • The large black hole is associated with a chemical potential µ which can only assume neg-ative values. This means that the fugacity z always lies between 0 and 1. We have shownthat in the limit z →
0, the Ruppeiner curvature vanishes indicating non-interacting be-haviour among the microstructures which is consistent with the interpretation that it isa classical limit for the large black hole. • The large black hole is associated with attractive interactions among the microstructures(
R <
0) in general away from the quantum limit. In the past, it has been argued in thecontext of quantum gases, that the divergence of Ruppeiner curvature could indicate aninstability among the microstructures and possibly correspond to Bose condensation. Inthe context of black holes in AdS, the limit z → R takes increasinglynegative values, if one considers the limit z →
1, indicating the dominance of long rangecorrelations and quantum mechanical fluctuations similar to the case of an ideal gas ofBosons. However, since no divergence of R (and hence ξ ) is observed where z ≈
1, itseems more appropriate to understand the z → • The small black hole branch is associated with both attraction and repulsion dominatedregions approximately separated by a second order critical point at which the specificheat C µ diverges. • Around the second order critical point of the small black hole, both the specific heatand the Ruppeiner curvature (hence, the correlation length) exhibit scaling behaviours, C µ ∼ | t | − and R ∼ | t | − where t is the reduced temperature.It would further be interesting to see the effects of charge on the black holes as well as highercurvature terms in the action on the microstructures of the large and small black holes in thisset up, although we can expect that some qualitative features shall remain the same as discussedin this paper. Acknowledgements
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