We study the thermodynamics of higher-dimensional singly spinning
asymptotically AdS black holes in the canonical (fixed J) ensemble of extended
phase space, where the cosmological constant is treated as pressure and the
corresponding conjugate quantity is interpreted as thermodynamic volume. Along
with the usual small/large black hole phase transition, we find a new
phenomenon of reentrant phase transitions for all d>5 dimensions, in which a
monotonic variation of the temperature yields two phase transitions from large
to small and back to large black holes. This situation is similar to that seen
in multicomponent liquids.Comment: 5 pages, 6 figure

Altamirano, Natacha

Kubiznak, David

Mann, Robert B.

Physical Review D

English

arXiv

We study the thermodynamics of higher-dimensional singly spinning asymptotically AdS black holes in the canonical (fixed J) ensemble of extended phase space, where the cosmological constant is treated as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. Along with the usual small/large black hole phase transition, we find a new phenomenon of reentrant phase transitions for all d ≥ 6 dimensions, in which a monotonic variation of the temperature yields two phase transitions from large to small and back to large black holes. This situation is similar to that seen in multicomponent liquids.Fil: Altamirano, Natacha Belén. University of Waterloo; Canadá. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; ArgentinaFil: Kubiznák, David. No especifíca;Fil: Mann, Robert B.. University of Waterloo; Canad

Altamirano, Natacha Belén

Kubiznák, David

CONICET Digital

arXiv:1306.5756v1  [hep-th]  24 Jun 2013pi-stronggrv-332Reentrant Phase Transitions in Rotating AdS Black HolesNatacha Altamirano,1, 2, ∗ David Kubiznˇa´k,1, † and Robert B. Mann1, 2, ‡1Perimeter Institute, 31 Caroline St. N. Waterloo Ontario, N2L 2Y5, Canada2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1(Dated: June 24, 2013)We study the thermodynamics of higher-dimensional singly spinning asymptotically AdS blackholes in the canonical (fixed J) ensemble of extended phase space, where the cosmological constantis treated as pressure and the corresponding conjugate quantity is interpreted as thermodynamicvolume. Along with the usual small/large black hole phase transition, we find a new phenomenonof reentrant phase transitions for all d ≥ 6 dimensions, in which a monotonic variation of thetemperature yields two phase transitions from large to small and back to large black holes. Thissituation is similar to that seen in multicomponent liquids.PACS numbers: 04.50.Gh, 04.70.-s, 05.70.CeIntroduction. In view of the AdS/CFT correspon-dence, phase transitions in asymptotically AdS blackholes allow for a dual interpretation in the ther-mal conformal field theory (CFT) living on the AdSboundary—the principal example being the well knownradiation/Schwarzschild-AdS black hole Hawking–Pagetransition [1] which can be interpreted as a confine-ment/deconfinement phase transition in the dual quarkgluon plasma [2]. Charged [3–6] and rotating [7, 8]asymptotically AdS back holes possess an interest-ing feature—they allow for a first order small-black-hole/large-black-hole phase (SBH/LBH) transition whichis in many ways reminiscent of the liquid/gas transitionof the Van der Waals fluid. This superficial analogy wasrecently found more intriguing [9] by considering a ther-modynamic analysis in an extended phase space wherethe cosmological constant is identified with thermody-namic pressure and its variations are included in the firstlaw of black hole thermodynamics. This notion emergesfrom geometric derivations of the Smarr formula [10] thati) imply the mass of an AdS black hole should be inter-preted as the enthalpy of the spacetime and ii) allow fora computation of the conjugate thermodynamic volume.Intensive and extensive quantities are now properly iden-tified [9] and the SBH/LBH transition can be understoodas a liquid/gas phase transition by employing Maxwell’sequal area law to the P − V diagram. Coexistence linesand critical exponents are then seen to match those of aVan der Waals fluid.In this paper we report the finding of an interestingphenomena, observed previously in multicomponent flu-ids, e.g., [11], of black hole reentrant phase transitions(RPTs). A system undergoes an RPT if a monotonicvariation of any thermodynamic quantity results in two(or more) phase transitions such that the final state ismacroscopically similar to the initial state. We find for acertain range of pressures (and a given angular momen-tum) that a monotonic lowering of the temperature yieldsa large-small-large black hole transition, where we referto the latter ‘large’ state as an intermediate black hole(IBH). This situation is accompanied by a discontinuityin the global minimum of the Gibbs free energy, referredto as a zeroth-order phase transition, a phenomenon seenin superfluidity and superconductivity [12], and recentlyfor Born–Infeld black holes [13]. We find the RPT to begeneric for all rotating AdS black holes in d ≥ 6 dimen-sions.Extended phase space thermodynamics. Rotating AdSblack holes were constructed in d = 4 by Carter [14]and later generalized to all higher dimensions [15–17].In what follows we limit ourselves to the case of singlyspinning black holes for which only one of the rotationparameters is non-trivial. In d spacetime dimensions themetric readsds2 = −∆ρ2(dt− aΞsin2θdϕ)2 +ρ2∆dr2 +ρ2Σdθ2+Σsin2θρ2[adt− (r2 + a2)Ξdϕ]2 + r2 cos2θdΩ2d−2 , (1)where dΩ2d−2 is the metric for the (d− 2)-sphere and∆ = (r2 + a2)(1 +r2l2)− 2mr5−d , Σ = 1− a2l2cos2θ ,Ξ = 1− a2l2, ρ2 = r2 + a2 cos2θ , (2)with l the AdS radius. The associated thermodynamicquantities read (in Planck units) [17]M =ωd−24πmΞ2(1 +(d− 4)Ξ2), (3)J =ωd−24πmaΞ2, ΩH =al2r2+ + l2r2+ + a2, (4)T =12π[r+(r2+l2+ 1)( 1a2 + r2++d− 32r2+)− 1r+], (5)S =ωd−24(a2 + r2+)rd−4+Ξ=A4, (6)where r+ is the black hole horizon radius (the largestpositive real root of ∆ = 0) and ωd = 2πd+12 /Γ[(d+1)/2]is the volume of the unit d-sphere.2We interpret the negative cosmological constant Λ asa positive thermodynamic pressure P [7, 10, 18, 19]P = − 18πΛ =(d− 1)(d− 2)16πl2, (7)in which case the first law of black hole thermodynamicsand Smarr formula [10] areδM = TδS +ΩHδJ + V δP , (8)d− 3d− 2M = TS +ΩHJ −2d− 2V P , (9)where the thermodynamic volume conjugate to P is [18]V =r+Ad− 1[1 +a2Ξ1 + r2+/l2(d− 2)r2+]. (10)It obeys the so called reverse isoperimetric inequality [18]and in the non-rotating case becomes V =ωd−2rd−1+d−1 ,which is the spatial volume of a round sphere of radiusr+ in the Euclidean space.Reentrant phase transition. The thermodynamic be-havior of the system is governed by the Gibbs free energyG = G(T, P, . . . ), which reads (I being the Euclidean ac-tion [17])G=M − TS = Iβ+ΩHJ (11)=ωd−2rd−5+16πΞ2(3a2 + r2+ −(r2+−a2)2l2+3a2r4+ + a4r2+l4)and depends on an external parameter J . We can plot itfor fixed J parametrically, first expressing a = a(J, r+, P )using (4) and then inserting the (well-behaved) solutioninto the expressions for G and T , consequently expressedas functions of P and r+.A simple criterion used for investigating the thermody-namic stability is the positivity of the specific heat. Fora canonical ensemble in the extended phase space, it isnatural to consider the specific heat at constant pressureCP = T(dSdT)P(12)which is different from the specific heat at constant ther-modynamic volume, CV . We take negativity of CP asa sign of local thermodynamic instability. Note that, asalways, we calculate this quantity for fixed J . That is,our specific heat at constant P is in fact a specific heatat constant (P, J) and coincides with CJ considered inprevious studies, e.g. [20]. When plotting the Gibbs freeenergy we plot branches with CP > 0 in red solid linesand branches with CP < 0 in dashed blue lines.The behavior of G depends crucially on the dimensiond. For d = 4 and d = 5 the situation is illustrated infig. 1. For P > Pc and any temperature there is only onebranch of locally thermodynamically stable black holes(with positive CP ) whereas for P < Pc the characteris-tic swallowtail behaviour indicating the SBH/LBH phasetransition emerges, with the global minimum of G havingCP > 0. The corresponding P − T diagram (not shown)is reminiscent of what was observed for charged blackholes in [9] and is analogous to the Van der Waals P −Tdiagram.In d ≥ 6 the situation is more subtle and markedlydifferent as shown in Figs. 2 and 3. For P > Pc, Gresembles the curve characteristic for the Schwarzschild-AdS black hole, known from the Hawking–Page transi-tion [1]: the upper branch corresponds to small unstableblack holes with CP < 0 whereas the lower branch de-scribes stable large black holes with CP > 0. There is acritical point at P = Pc, and for the range of pressuresP ∈ (Pt, Pc) and temperatures T ∈ (Tt, Tc) there is astandard first order SBH/LBH phase transition, reminis-cent of the Van der Waals phase transition [9]. Howeverfor Pc > Pz = P > Pt three separate phases of blackholes emerge (see fig. 4): intermediate black holes (IBH)(on the left), small (middle), and large (on the right).This holds for T ∈ (Tt, Tz), P ∈ (Pt, Pz) and terminatesat T = Tt. Small and large black holes are separated by astandard first order phase transition, but the intermedi-ate and small are separated by a finite jump in G, whichin this range has a discontinuous global minimum (fig. 3).This is the RPT, first observed in a nicotine/water mix-ture [21], and since seen in multicomponent fluid systems,gels, ferroelectrics, liquid crystals, and binary gases [11].Finally, for T < Tt only one LBH phase exists.Consider fig. 3. If we start decreasing the temperaturefrom, say T = 0.24, the system follows the lower solid(red) curve until it joins the upper solid (red) curve—this corresponds to a first order SBH/LBH phase tran-sition. As T continues to decrease the system followsthis upper curve until T = T0 ∈ (Tt, Tz), where G hasa discontinuity at its global minimum. Further decreas-ing T , the system jumps to the uppermost red line—thiscorresponds to the zeroth order phase transition betweensmall and intermediate black holes.This novel situation is clearly illustrated in the P − Tdiagrams in fig. 4. There is the expected SBH/LBH lineof coexistence corresponding to the liquid/gas Van derWalls case, ending in a critical point (Tc, Pc). This lineterminates at (Tt, Pt), where there is a “triple point” be-tween the small, intermediate, and large black holes. Forsmaller values of T there is an unstable line of coexis-tence (not shown) between the IBHs and LBHs. For T ∈(Tt, Tz) there is a new IBH/SBH line of coexistence (seeinset of fig. 4) that terminates in another “critical point”(Tz, Pz). The range for the RPT is quite narrow and mustbe determined numerically. For example for J = 1 andd = 6 we obtain (Tt, Tz, Tc) ≈ (0.2332, 0.2349, 0.3004)and (Pt, Pz, Pc) ≈ (0.0553, 0.0579, 0.0958).Equation of state. The equation of state P =P (V, T, J) in the canonical (fixed J) ensemble can3P<Pc P=Pc P>PcGT0.20.1 0.150.20.611.41.8FIG. 1. Gibbs free energy in d = 5 for various valuesof P and J = 1. As with RN-AdS black holes in any dimen-sion, we see characteristic swallowtail behaviour indicatingan SBH/LBH transition. Solid-red/dashed-blue lines corre-spond to CP positive/negative respectively; the CP < 0 lineindicates a local thermodynamic instability where the Gibbsenergy is not a local minimum; at the joins CP diverges. Thebehaviour of G in d = 4 is similar.P <Pc P =PcP >PcGTP t < P <P z0.2 0.25 0.3 0.350.611.41.8FIG. 2. Gibbs free energy in d = 6 for various values ofP and J = 1. Solid-red/dashed-blue lines correspond to CPpositive/negative respectively. As with Schwarzschild-AdSblack holes, for P ≥ Pc, the (lower) LBH branch is thermody-namically stable whereas the upper branch is unstable. ForP = Pc we observe critical behaviour. At the joins of dashed(blue) and solid (red) lines CP diverges. For P ∈ (Pt, Pz) weobserve a “zeroth-order phase transition” signifying the onsetof an RPT.be computed by solving Eqs. (3)—(6) to eliminate(m, a, r+) in terms of the basic thermodynamic variables.The result can be obtained numerically (see fig. 5), butthere are two cases of physical interest that can be ap-proximated analytically: the slowly rotating (a→ 0) caseand the ultraspinning (a→ l) regime.GTT 00.24 0.2611.52FIG. 3. Zeroth order phase transition in d = 6. Aclose-up of fig. 2 illustrating the discontinuity in the globalminimum of G at T = T0 ≈ 0.2339 ∈ (Tt, Tz) (denoted bythe vertical line). We have set P = 0.0564 ∈ (Pt, Pz) andJ = 1. Solid-red/dashed-blue lines correspond to CP posi-tive/negative respectively.FIG. 4. P − T diagram in d = 6. The coexistence line ofthe first order phase transition between small and large blackholes is depicted by a thick black solid line for J = 1. It initi-ates from the critical point (Pc, Tc) and terminates at (Pt, Tt).The red solid line in the inset indicates the ‘coexistence line’of small and intermediate black holes, separated by a finitegap in G, indicating the RPT. It commences from (Tz, Pz)and terminates at (Pt, Tt). A similar figure is valid for anyd ≥ 6.In the slowing rotating case, we expand in the param-eter ǫ = a/l, to obtainP =Tv− d− 3π(d− 2)v2+π(d − 1)16dJ24ω2d−2[(d−2)v]2(d−1)+O(ǫ4) (13)and it is straightforward to show that Van der Waals4behaviour occurs as in d = 4 [13]. The specific volume ofthe fluid v is defined by V = (κv)d−1ωd−2d−1 , κ =14 (d − 2).Critical points (Pc, vc, Tc) can be computed from∂P∂v=∂2P∂v2= 0. We find that the critical exponents α = 0,β = 1/2, γ = 1, δ = 3 match those of a Van der Waalsfluid, though the critical ratio ρc =PcvcTc= 2d−34(d−1) differs;note that it reduces to ρc = 5/12 for d = 4 [13].In the ultraspinning limit a→ l, r+ → 0. For all d ≥ 6the geometry of a black hole approaches that of a blackmembrane [22, 23]; setting f = 1− µR5−d the metric isds2M = −fdτ2 +dR2f+ dσ2 + σ2dϕ2 +R2dΩ2d−4 ,known to be unstable due to the Gregory–Laflamme in-stability [24]. The entropy vanishes and the temperaturediverges but both the angular momentum and (specific)volume V = 8piMl2(d−1)(d−2) remain finite, while the equationof state is P = 4pi(d−1)(d−2)J2V 2. The same equation of stateis valid for ultraspinning black rings [25]. Expandingabout the ultraspinning limit, Ξ→ 0, we findP6 =πJ25V 2−√5ω416√π3V T+9√5ω4J23200√π3V 5T 5+O(Ξ9) ,G6 =2√5πPJ5+√25ω4J8√T (πP )14− 5ω41024π3TP+O(Ξ3)for d = 6; higher-dimensional expansions can likewise becomputed.In between these two limiting cases a phase transitionto a novel family of black objects branching off the spher-ical black holes (e.g., black rings) is expected (see, e.g.,[26, 27] and references therein). The exact critical pointis shifted; we find numerically that the critical ratio ρc isslightly smaller and that all critical exponents remain thesame. The two expansions for the equation of state are,together with the exact numerical solution, displayed forthe critical temperature in fig. 6. Obviously, the slowrotation approximation is effectively accurate for largevalues of r+ (rightmost curve) whereas the ultraspinningone (leftmost curve) is accurate as r+ approaches zero.Discussion. Although the cosmological constant isgenerally regarded as fixed in the action, it is possibleto dynamically generate it using a (d − 1)-form gaugepotential and incorporate it into a generalized first-law[28]. We see that identifying it as pressure and incorpo-rating its conjugate volume yields not only a consistentSmarr formula, but also a qualitatively new phase struc-ture in the thermodynamics of rotating black holes simi-lar to binary fluids. In binary fluids at low temperatures,directional bonding between unlike species can lead toa miscible state, which is restored at high temperaturessince entropy of mixing dominates. At intermediate tem-peratures the two fluids become immiscible. The corre-sponding physics for d ≥ 6 rotating black holes remainsan interesting subject for further study.FIG. 5. P − v diagram in d = 5 (inset) and d = 6. Obvi-ously, in d ≥ 6 the P − v diagram is more complex than thatof the standard Van der Waals, and reflects the interestingbehavior of the Gibbs free energy and a possible RPT.ULTRASPINSLOWT<TcT=TcEXACTvP00.050.10.150.20.5 1 1.5 2 2.5 3FIG. 6. The two approximations. Exact critical and sub-critical isotherms, depicted by black crosses, are compared tothe slow spinning expansion (13) denoted by red curves andthe ultraspinning expansion (14) denoted by a blue curve.Note that the ultraspinning black holes correspond to the up-per branch in the Gibbs free energy which is unstable. Wehave considered d = 6 and set J = 1.Acknowledgments. This work was supported in part bythe Natural Sciences and Engineering Research Councilof Canada. We are grateful to the Perimeter Institutewhere part of this work was carried out.∗ naltamirano@perimeterinstitute.ca† dkubiznak@perimeterinstitute.ca‡ rbmann@uwaterloo.ca5[1] S. Hawking and D. N. Page, Thermodynamics of BlackHoles in anti-De Sitter Space, Commun.Math.Phys. 87(1983) 577.[2] E. Witten, Anti-de Sitter space, thermal phasetransition, and confinement in gauge theories,Adv.Theor.Math.Phys. 2 (1998) 505–532,[hep-th/9803131].[3] A. Chamblin, R. Emparan, C. Johnson, and R. Myers,Charged AdS black holes and catastrophic holography,Phys.Rev. D60 (1999) 064018, [hep-th/9902170].[4] A. Chamblin, R. Emparan, C. Johnson, and R. Myers,Holography, thermodynamics and fluctuations of chargedAdS black holes, Phys.Rev. D60 (1999) 104026,[hep-th/9904197].[5] M. Cvetic and S. Gubser, Phases of R charged blackholes, spinning branes and strongly coupled gaugetheories, JHEP 9904 (1999) 024, [hep-th/9902195].[6] C. V. Johnson, Large N Phase Transitions, FiniteVolume, and Entanglement Entropy, arXiv:1306.4955.[7] M. M. Caldarelli, G. Cognola, and D. Klemm,Thermodynamics of Kerr-Newman-AdS black holes andconformal field theories, Class.Quant.Grav. 17 (2000)399–420, [hep-th/9908022].[8] Y.-D. Tsai, X. Wu, and Y. Yang, Phase Structure ofKerr-AdS Black Hole, Phys.Rev. D85 (2012) 044005,[arXiv:1104.0502].[9] D. Kubiznak and R. B. Mann, P-V criticality of chargedAdS black holes, JHEP 1207 (2012) 033,[arXiv:1205.0559].[10] D. Kastor, S. Ray, and J. Traschen, Enthalpy and theMechanics of AdS Black Holes, Class.Quant.Grav. 26(2009) 195011, [arXiv:0904.2765].[11] T. Narayanan and A. Kumar, Reentrant phasetransitions in multicomponent liquid mixtures, PhysicsReports 249 (1994) 135–218.[12] V. P. Maslov, Zeroth-order phase transitions,Mathematical Notes 76 (2004), no. 5-6 697–710.[13] S. Gunasekaran, D. Kubiznak, and R. Mann, Extendedphase space thermodynamics for charged and rotatingblack holes and Born-Infeld vacuum polarization, JHEP1211 (2012) 110, [arXiv:1208.6251].[14] B. Carter, Hamilton-Jacobi and Schrodinger separablesolutions of Einstein’s equations, Commun. Math. Phys.10 (1968) 280–310.[15] S. W. Hawking, C. J. Hunter, and M. M.Taylor-Robinson, Rotation and the AdS/CFTcorrespondence, Phys. Rev. D59 (1999) 064005,[hep-th/9811056].[16] G. W. Gibbons, H. Lu¨, D. N. Page, and C. N. Pope,Rotating black holes in higher dimensions with acosmological constant, Phys. Rev. Lett. 93 (2004)171102, [hep-th/0409155].[17] G. W. Gibbons, H. Lu¨, D. N. Page, and C. N. Pope,The general Kerr-de Sitter metrics in all dimensions, J.Geom. Phys. 53 (2005) 49–73, [hep-th/0404008].[18] M. Cvetic, G. Gibbons, D. Kubiznak, and C. Pope,Black Hole Enthalpy and an Entropy Inequality for theThermodynamic Volume, Phys.Rev. D84 (2011) 024037,[arXiv:1012.2888].[19] B. P. Dolan, Where is the PdV term in the fist law ofblack hole thermodynamics?, arXiv:1209.1272.[20] R. Monteiro, M. J. Perry, and J. E. Santos,Thermodynamic instability of rotating black holes,Phys.Rev. D80 (2009) 024041, [arXiv:0903.3256].[21] C. Hudson Z. Phys. Chem. 47 (1904) 113.[22] R. Emparan and R. C. Myers, Instability ofultra-spinning black holes, JHEP 0309 (2003) 025,[hep-th/0308056].[23] M. Caldarelli, R. Emparan, and M. J. Rodriguez, BlackRings in (Anti)-deSitter space, JHEP 0811 (2008) 011,[arXiv:0806.1954].[24] R. Gregory and R. Laflamme, Black strings andp-branes are unstable, Phys.Rev.Lett. 70 (1993)2837–2840, [hep-th/9301052].[25] N. Altamirano, D., D. Kubiznak, and R. Mann,Thermodynamic volume, criticality, and theultraspinning limit of higher-dimensional black holes,in preparation.[26] O. Dias, P. Figueras, R. Monteiro, and J. Santos,Ultraspinning instability of anti-de Sitter black holes,JHEP 1012 (2010) 067, [1011.0996].[27] G. S. Hartnett and J. E. Santos, Non-AxisymmetricInstability of Rotating Black Holes in HigherDimensions, arXiv:1306.4318.[28] J. Creighton and R. B. Mann, Quasilocalthermodynamics of dilaton gravity coupled to gaugefields, Phys.Rev. D52 (1995) 4569–4587,[gr-qc/9505007].

Reentrant Phase Transitions in Rotating AdS Black Holes

Natacha Altamirano

David Kubizňák

Robert B. Mann

Crossref

Reentrant phase transitions in rotating anti–de Sitter black holes

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:1306.5756v1  [hep-th]  24 Jun 2013pi-stronggrv-332Reentrant Phase Transitions in Rotating AdS Black HolesNatacha Altamirano,1, 2, ∗ David Kubizňák,1, † and Robert B. Mann1, 2, ‡1Perimeter Institute, 31 Caroline St. N. Waterloo Ontario, N2L 2Y5, Canada2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1(Dated: June 24, 2013)We study the thermodynamics of higher-dimensional singly spinning asymptotically AdS blackholes in the canonical (fixed J) ensemble of extended phase space, where the cosmological constantis treated as pressure and the corresponding conjugate quantity is interpreted as thermodynamicvolume. Along with the usual small/large black hole phase transition, we find a new phenomenonof reentrant phase transitions for all d ≥ 6 dimensions, in which a monotonic variation of thetemperature yields two phase transitions from large to small and back to large black holes. Thissituation is similar to that seen in multicomponent liquids.PACS numbers: 04.50.Gh, 04.70.-s, 05.70.CeIntroduction. In view of the AdS/CFT correspon-dence, phase transitions in asymptotically AdS blackholes allow for a dual interpretation in the ther-mal conformal field theory (CFT) living on the AdSboundary—the principal example being the well knownradiation/Schwarzschild-AdS black hole Hawking–Pagetransition [1] which can be interpreted as a confine-ment/deconfinement phase transition in the dual quarkgluon plasma [2]. Charged [3–6] and rotating [7, 8]asymptotically AdS back holes possess an interest-ing feature—they allow for a first order small-black-hole/large-black-hole phase (SBH/LBH) transition whichis in many ways reminiscent of the liquid/gas transitionof the Van der Waals fluid. This superficial analogy wasrecently found more intriguing [9] by considering a ther-modynamic analysis in an extended phase space wherethe cosmological constant is identified with thermody-namic pressure and its variations are included in the firstlaw of black hole thermodynamics. This notion emergesfrom geometric derivations of the Smarr formula [10] thati) imply the mass of an AdS black hole should be inter-preted as the enthalpy of the spacetime and ii) allow fora computation of the conjugate thermodynamic volume.Intensive and extensive quantities are now properly iden-tified [9] and the SBH/LBH transition can be understoodas a liquid/gas phase transition by employing Maxwell’sequal area law to the P − V diagram. Coexistence linesand critical exponents are then seen to match those of aVan der Waals fluid.In this paper we report the finding of an interestingphenomena, observed previously in multicomponent flu-ids, e.g., [11], of black hole reentrant phase transitions(RPTs). A system undergoes an RPT if a monotonicvariation of any thermodynamic quantity results in two(or more) phase transitions such that the final state ismacroscopically similar to the initial state. We find for acertain range of pressures (and a given angular momen-tum) that a monotonic lowering of the temperature yieldsa large-small-large black hole transition, where we referto the latter ‘large’ state as an intermediate black hole(IBH). This situation is accompanied by a discontinuityin the global minimum of the Gibbs free energy, referredto as a zeroth-order phase transition, a phenomenon seenin superfluidity and superconductivity [12], and recentlyfor Born–Infeld black holes [13]. We find the RPT to begeneric for all rotating AdS black holes in d ≥ 6 dimen-sions.Extended phase space thermodynamics. Rotating AdSblack holes were constructed in d = 4 by Carter [14]and later generalized to all higher dimensions [15–17].In what follows we limit ourselves to the case of singlyspinning black holes for which only one of the rotationparameters is non-trivial. In d spacetime dimensions themetric readsds2 = −∆ρ2(dt−aΞsin2θdϕ)2 +ρ2∆dr2 +ρ2Σdθ2+Σsin2θρ2[adt−(r2 + a2)Ξdϕ]2 + r2 cos2θdΩ2d−2 , (1)where dΩ2d−2 is the metric for the (d− 2)-sphere and∆ = (r2 + a2)(1 +r2l2)− 2mr5−d , Σ = 1−a2l2cos2θ ,Ξ = 1−a2l2, ρ2 = r2 + a2 cos2θ , (2)with l the AdS radius. The associated thermodynamicquantities read (in Planck units) [17]M =ωd−24πmΞ2(1 +(d− 4)Ξ2), (3)J =ωd−24πmaΞ2, ΩH =al2r2+ + l2r2+ + a2, (4)T =12π[r+(r2+l2+ 1)(1a2 + r2++d− 32r2+)−1r+], (5)S =ωd−24(a2 + r2+)rd−4+Ξ=A4, (6)where r+ is the black hole horizon radius (the largestpositive real root of ∆ = 0) and ωd = 2πd+12 /Γ[(d+1)/2]is the volume of the unit d-sphere.2We interpret the negative cosmological constant Λ asa positive thermodynamic pressure P [7, 10, 18, 19]P = −18πΛ =(d− 1)(d− 2)16πl2, (7)in which case the first law of black hole thermodynamicsand Smarr formula [10] areδM = TδS +ΩHδJ + V δP , (8)d− 3d− 2M = TS +ΩHJ −2d− 2V P , (9)where the thermodynamic volume conjugate to P is [18]V =r+Ad− 1[1 +a2Ξ1 + r2+/l2(d− 2)r2+]. (10)It obeys the so called reverse isoperimetric inequality [18]and in the non-rotating case becomes V =ωd−2rd−1+d−1 ,which is the spatial volume of a round sphere of radiusr+ in the Euclidean space.Reentrant phase transition. The thermodynamic be-havior of the system is governed by the Gibbs free energyG = G(T, P, . . . ), which reads (I being the Euclidean ac-tion [17])G=M − TS =Iβ+ΩHJ (11)=ωd−2rd−5+16πΞ2(3a2 + r2+ −(r2+−a2)2l2+3a2r4+ + a4r2+l4)and depends on an external parameter J . We can plot itfor fixed J parametrically, first expressing a = a(J, r+, P )using (4) and then inserting the (well-behaved) solutioninto the expressions for G and T , consequently expressedas functions of P and r+.A simple criterion used for investigating the thermody-namic stability is the positivity of the specific heat. Fora canonical ensemble in the extended phase space, it isnatural to consider the specific heat at constant pressureCP = T(dSdT)P(12)which is different from the specific heat at constant ther-modynamic volume, CV . We take negativity of CP asa sign of local thermodynamic instability. Note that, asalways, we calculate this quantity for fixed J . That is,our specific heat at constant P is in fact a specific heatat constant (P, J) and coincides with CJ considered inprevious studies, e.g. [20]. When plotting the Gibbs freeenergy we plot branches with CP > 0 in red solid linesand branches with CP < 0 in dashed blue lines.The behavior of G depends crucially on the dimensiond. For d = 4 and d = 5 the situation is illustrated infig. 1. For P > Pc and any temperature there is only onebranch of locally thermodynamically stable black holes(with positive CP ) whereas for P < Pc the characteris-tic swallowtail behaviour indicating the SBH/LBH phasetransition emerges, with the global minimum of G havingCP > 0. The corresponding P − T diagram (not shown)is reminiscent of what was observed for charged blackholes in [9] and is analogous to the Van der Waals P −Tdiagram.In d ≥ 6 the situation is more subtle and markedlydifferent as shown in Figs. 2 and 3. For P > Pc, Gresembles the curve characteristic for the Schwarzschild-AdS black hole, known from the Hawking–Page transi-tion [1]: the upper branch corresponds to small unstableblack holes with CP < 0 whereas the lower branch de-scribes stable large black holes with CP > 0. There is acritical point at P = Pc, and for the range of pressuresP ∈ (Pt, Pc) and temperatures T ∈ (Tt, Tc) there is astandard first order SBH/LBH phase transition, reminis-cent of the Van der Waals phase transition [9]. Howeverfor Pc > Pz = P > Pt three separate phases of blackholes emerge (see fig. 4): intermediate black holes (IBH)(on the left), small (middle), and large (on the right).This holds for T ∈ (Tt, Tz), P ∈ (Pt, Pz) and terminatesat T = Tt. Small and large black holes are separated by astandard first order phase transition, but the intermedi-ate and small are separated by a finite jump in G, whichin this range has a discontinuous global minimum (fig. 3).This is the RPT, first observed in a nicotine/water mix-ture [21], and since seen in multicomponent fluid systems,gels, ferroelectrics, liquid crystals, and binary gases [11].Finally, for T < Tt only one LBH phase exists.Consider fig. 3. If we start decreasing the temperaturefrom, say T = 0.24, the system follows the lower solid(red) curve until it joins the upper solid (red) curve—this corresponds to a first order SBH/LBH phase tran-sition. As T continues to decrease the system followsthis upper curve until T = T0 ∈ (Tt, Tz), where G hasa discontinuity at its global minimum. Further decreas-ing T , the system jumps to the uppermost red line—thiscorresponds to the zeroth order phase transition betweensmall and intermediate black holes.This novel situation is clearly illustrated in the P − Tdiagrams in fig. 4. There is the expected SBH/LBH lineof coexistence corresponding to the liquid/gas Van derWalls case, ending in a critical point (Tc, Pc). This lineterminates at (Tt, Pt), where there is a “triple point” be-tween the small, intermediate, and large black holes. Forsmaller values of T there is an unstable line of coexis-tence (not shown) between the IBHs and LBHs. For T ∈(Tt, Tz) there is a new IBH/SBH line of coexistence (seeinset of fig. 4) that terminates in another “critical point”(Tz, Pz). The range for the RPT is quite narrow and mustbe determined numerically. For example for J = 1 andd = 6 we obtain (Tt, Tz, Tc) ≈ (0.2332, 0.2349, 0.3004)and (Pt, Pz, Pc) ≈ (0.0553, 0.0579, 0.0958).Equation of state. The equation of state P =P (V, T, J) in the canonical (fixed J) ensemble can3P<Pc P=Pc P>PcGT0.20.1 0.150.20.611.41.8FIG. 1. Gibbs free energy in d = 5 for various valuesof P and J = 1. As with RN-AdS black holes in any dimen-sion, we see characteristic swallowtail behaviour indicatingan SBH/LBH transition. Solid-red/dashed-blue lines corre-spond to CP positive/negative respectively; the CP < 0 lineindicates a local thermodynamic instability where the Gibbsenergy is not a local minimum; at the joins CP diverges. Thebehaviour of G in d = 4 is similar.P <Pc P =PcP >PcGTP t < P <P z0.2 0.25 0.3 0.350.611.41.8FIG. 2. Gibbs free energy in d = 6 for various values ofP and J = 1. Solid-red/dashed-blue lines correspond to CPpositive/negative respectively. As with Schwarzschild-AdSblack holes, for P ≥ Pc, the (lower) LBH branch is thermody-namically stable whereas the upper branch is unstable. ForP = Pc we observe critical behaviour. At the joins of dashed(blue) and solid (red) lines CP diverges. For P ∈ (Pt, Pz) weobserve a “zeroth-order phase transition” signifying the onsetof an RPT.be computed by solving Eqs. (3)—(6) to eliminate(m, a, r+) in terms of the basic thermodynamic variables.The result can be obtained numerically (see fig. 5), butthere are two cases of physical interest that can be ap-proximated analytically: the slowly rotating (a → 0) caseand the ultraspinning (a → l) regime.GTT 00.24 0.2611.52FIG. 3. Zeroth order phase transition in d = 6. Aclose-up of fig. 2 illustrating the discontinuity in the globalminimum of G at T = T0 ≈ 0.2339 ∈ (Tt, Tz) (denoted bythe vertical line). We have set P = 0.0564 ∈ (Pt, Pz) andJ = 1. Solid-red/dashed-blue lines correspond to CP posi-tive/negative respectively.FIG. 4. P − T diagram in d = 6. The coexistence line ofthe first order phase transition between small and large blackholes is depicted by a thick black solid line for J = 1. It initi-ates from the critical point (Pc, Tc) and terminates at (Pt, Tt).The red solid line in the inset indicates the ‘coexistence line’of small and intermediate black holes, separated by a finitegap in G, indicating the RPT. It commences from (Tz, Pz)and terminates at (Pt, Tt). A similar figure is valid for anyd ≥ 6.In the slowing rotating case, we expand in the param-eter ǫ = a/l, to obtainP =Tv−d− 3π(d− 2)v2+π(d − 1)16dJ24ω2d−2[(d−2)v]2(d−1)+O(ǫ4) (13)and it is straightforward to show that Van der Waals4behaviour occurs as in d = 4 [13]. The specific volume ofthe fluid v is defined by V = (κv)d−1ωd−2d−1 , κ =14 (d − 2).Critical points (Pc, vc, Tc) can be computed from∂P∂v=∂2P∂v2= 0. We find that the critical exponents α = 0,β = 1/2, γ = 1, δ = 3 match those of a Van der Waalsfluid, though the critical ratio ρc =PcvcTc= 2d−34(d−1) differs;note that it reduces to ρc = 5/12 for d = 4 [13].In the ultraspinning limit a → l, r+ → 0. For all d ≥ 6the geometry of a black hole approaches that of a blackmembrane [22, 23]; setting f = 1− µR5−d the metric isds2M = −fdτ2 +dR2f+ dσ2 + σ2dϕ2 +R2dΩ2d−4 ,known to be unstable due to the Gregory–Laflamme in-stability [24]. The entropy vanishes and the temperaturediverges but both the angular momentum and (specific)volume V = 8πMl2(d−1)(d−2) remain finite, while the equationof state is P = 4π(d−1)(d−2)J2V 2. The same equation of stateis valid for ultraspinning black rings [25]. Expandingabout the ultraspinning limit, Ξ → 0, we findP6 =πJ25V 2−√5ω416√π3V T+9√5ω4J23200√π3V 5T 5+O(Ξ9) ,G6 =2√5πPJ5+√25ω4J8√T (πP )14−5ω41024π3TP+O(Ξ3)for d = 6; higher-dimensional expansions can likewise becomputed.In between these two limiting cases a phase transitionto a novel family of black objects branching off the spher-ical black holes (e.g., black rings) is expected (see, e.g.,[26, 27] and references therein). The exact critical pointis shifted; we find numerically that the critical ratio ρc isslightly smaller and that all critical exponents remain thesame. The two expansions for the equation of state are,together with the exact numerical solution, displayed forthe critical temperature in fig. 6. Obviously, the slowrotation approximation is effectively accurate for largevalues of r+ (rightmost curve) whereas the ultraspinningone (leftmost curve) is accurate as r+ approaches zero.Discussion. Although the cosmological constant isgenerally regarded as fixed in the action, it is possibleto dynamically generate it using a (d − 1)-form gaugepotential and incorporate it into a generalized first-law[28]. We see that identifying it as pressure and incorpo-rating its conjugate volume yields not only a consistentSmarr formula, but also a qualitatively new phase struc-ture in the thermodynamics of rotating black holes simi-lar to binary fluids. In binary fluids at low temperatures,directional bonding between unlike species can lead toa miscible state, which is restored at high temperaturessince entropy of mixing dominates. At intermediate tem-peratures the two fluids become immiscible. The corre-sponding physics for d ≥ 6 rotating black holes remainsan interesting subject for further study.FIG. 5. P − v diagram in d = 5 (inset) and d = 6. Obvi-ously, in d ≥ 6 the P − v diagram is more complex than thatof the standard Van der Waals, and reflects the interestingbehavior of the Gibbs free energy and a possible RPT.ULTRASPINSLOWT<TcT=TcEXACTvP00.050.10.150.20.5 1 1.5 2 2.5 3FIG. 6. The two approximations. Exact critical and sub-critical isotherms, depicted by black crosses, are compared tothe slow spinning expansion (13) denoted by red curves andthe ultraspinning expansion (14) denoted by a blue curve.Note that the ultraspinning black holes correspond to the up-per branch in the Gibbs free energy which is unstable. 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Reentrant Phase Transitions in Rotating AdS Black Holes

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