# Critical exponents of finite temperature chiral phase transition in soft-wall AdS/QCD models

- 18 Downloads

## Abstract

Criticality of chiral phase transition at finite temperature is investigated in a soft-wall AdS/QCD model, with two, three degenerate flavors (*N*_{f} = 2*,* 3) and two light plus one heavier flavor (*N*_{f} = 2 + 1). It is shown that in quark mass plane (*m*_{u/d} − *m*_{s}) chiral phase transition is second order at a certain critical line, by which the whole plane is divided into first order and crossover regions. The critical exponents *β* and *δ*, describing critical behavior of chiral condensate along temperature axis and light quark mass axis, are extracted both numerically and analytically. The model gives the critical exponents of the values \( \beta =\frac{1}{2},\delta =3 \) and \( \beta =\frac{1}{3},\delta =3 \) for *N*_{f} = 2 and *N*_{f} = 3 respectively. For *N*_{f} = 2 + 1, in small strange quark mass (*m*_{s}) region, the phase transitions for strange quark and *u/d* quarks are strongly coupled, and the critical exponents are \( \beta =\frac{1}{3},\delta =3 \); when *m*_{s} is larger than *m*_{s,t} = 0*.*290 GeV, the dynamics of light flavors (*u, d*) and strange quarks decoupled and the critical exponents for *ūu* and \( \overline{d}d \) becomes \( \beta =\frac{1}{2},\delta =3 \), exactly the same as *N*_{f} = 2 result and the mean field result of 3D Ising model; between the two segments, there is a tri-critical point at *m*_{s,t} = 0*.*290 GeV, at which \( \beta =\frac{1}{4},\delta =5 \). In some sense, the current results is still at mean field level, and we also showed the possibility to go beyond mean field approximation by including the higher power of scalar potential and the temperature dependence of dilaton field, which might be reasonable in a full back-reaction model. The current study might also provide reasonable constraints on constructing a realistic holographic QCD model, which could describe both chiral dynamics and glue-dynamics correctly.

## Keywords

Holography and quark-gluon plasmas AdS-CFT Correspondence Gaugegravity correspondence## Notes

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

- [1]R.D. Pisarski and F. Wilczek,
*Remarks on the Chiral Phase Transition in Chromodynamics*,*Phys. Rev.***D 29**(1984) 338 [INSPIRE]. - [2]T. Hatsuda and T. Kunihiro,
*Fluctuation Effects in Hot Quark Matter: Precursors of Chiral Transition at Finite Temperature*,*Phys. Rev. Lett.***55**(1985) 158 [INSPIRE].ADSCrossRefGoogle Scholar - [3]E. Laermann and O. Philipsen,
*The Status of lattice QCD at finite temperature*,*Ann. Rev. Nucl. Part. Sci.***53**(2003) 163 [hep-ph/0303042] [INSPIRE]. - [4]T. Schäfer and E.V. Shuryak,
*Instantons in QCD*,*Rev. Mod. Phys.***70**(1998) 323 [hep-ph/9610451] [INSPIRE]. - [5]M.A. Stephanov, K. Rajagopal and E.V. Shuryak,
*Signatures of the tricritical point in QCD*,*Phys. Rev. Lett.***81**(1998) 4816 [hep-ph/9806219] [INSPIRE]. - [6]F. Karsch, E. Laermann and A. Peikert,
*Quark mass and flavor dependence of the QCD phase transition*,*Nucl. Phys.***B 605**(2001) 579 [hep-lat/0012023] [INSPIRE]. - [7]N. Itoh,
*Hydrostatic Equilibrium of Hypothetical Quark Stars*,*Prog. Theor. Phys.***44**(1970) 291 [INSPIRE].ADSCrossRefGoogle Scholar - [8]R. Rapp and J. Wambach,
*Chiral symmetry restoration and dileptons in relativistic heavy ion collisions*,*Adv. Nucl. Phys.***25**(2000) 1 [hep-ph/9909229] [INSPIRE]. - [9]A. Palmese, W. Cassing, E. Seifert, T. Steinert, P. Moreau and E.L. Bratkovskaya,
*Chiral symmetry restoration in heavy-ion collisions at intermediate energies*,*Phys. Rev.***C 94**(2016) 044912 [arXiv:1607.04073] [INSPIRE]. - [10]J. Schukraft and R. Stock,
*Toward the Limits of Matter: Ultra-relativistic nuclear collisions at CERN*,*Adv. Ser. Direct. High Energy Phys.***23**(2015) 61 [arXiv:1505.06853] [INSPIRE].ADSCrossRefGoogle Scholar - [11]F.R. Brown et al.,
*On the existence of a phase transition for QCD with three light quarks*,*Phys. Rev. Lett.***65**(1990) 2491 [INSPIRE].ADSCrossRefGoogle Scholar - [12]H.-T. Ding, F. Karsch and S. Mukherjee,
*Thermodynamics of strong-interaction matter from Lattice QCD*,*Int. J. Mod. Phys.***E 24**(2015) 1530007 [arXiv:1504.05274] [INSPIRE]. - [13]G. Endrodi, Z. Fodor, S.D. Katz and K.K. Szabo,
*The Nature of the finite temperature QCD transition as a function of the quark masses*, PoS(LATTICE2007)182 (2007) [arXiv:0710.0998] [INSPIRE]. - [14]M.E. Fisher,
*The renormalization group in the theory of critical behavior*,*Rev. Mod. Phys.***46**(1974) 597 [*Erratum ibid.***47**(1975) 543] [INSPIRE]. - [15]P.C. Hohenberg and B.I. Halperin,
*Theory of Dynamic Critical Phenomena*,*Rev. Mod. Phys.***49**(1977) 435 [INSPIRE].ADSCrossRefGoogle Scholar - [16]Z. Fodor and S.D. Katz,
*Lattice determination of the critical point of QCD at finite T and mu*,*JHEP***03**(2002) 014 [hep-lat/0106002] [INSPIRE]. - [17]Z. Fodor and S.D. Katz,
*Critical point of QCD at finite T and mu, lattice results for physical quark masses*,*JHEP***04**(2004) 050 [hep-lat/0402006] [INSPIRE]. - [18]P. de Forcrand and O. Philipsen,
*The Chiral critical line of N*_{f}= 2 + 1*QCD at zero and non-zero baryon density*,*JHEP***01**(2007) 077 [hep-lat/0607017] [INSPIRE]. - [19]STAR collaboration,
*An Experimental Exploration of the QCD Phase Diagram: The Search for the Critical Point and the Onset of De-confinement*, arXiv:1007.2613 [INSPIRE]. - [20]G. Odyniec,
*RHIC Beam Energy Scan Program: Phase I and II*, PoS(CPOD2013)043 (2013) [INSPIRE]. - [21]X. Luo and N. Xu,
*Search for the QCD Critical Point with Fluctuations of Conserved Quantities in Relativistic Heavy-Ion Collisions at RHIC: An Overview*,*Nucl. Sci. Tech.***28**(2017) 112 [arXiv:1701.02105] [INSPIRE]. - [22]A. Pelissetto and E. Vicari,
*Critical phenomena and renormalization group theory*,*Phys. Rept.***368**(2002) 549 [cond-mat/0012164] [INSPIRE]. - [23]CP-PACS collaboration,
*Phase structure and critical temperature of two flavor QCD with renormalization group improved gauge action and clover improved Wilson quark action*,*Phys. Rev.***D 63**(2000) 034502 [hep-lat/0008011] [INSPIRE]. - [24]S. Ejiri et al.,
*On the magnetic equation of state in*(2 + 1)*-flavor QCD*,*Phys. Rev.***D 80**(2009) 094505 [arXiv:0909.5122] [INSPIRE]. - [25]F. Karsch,
*O(N) universality and the chiral phase transition in QCD*,*Prog. Theor. Phys. Suppl.***186**(2010) 479 [arXiv:1007.2393] [INSPIRE].ADSCrossRefGoogle Scholar - [26]O. Kaczmarek et al.,
*Phase boundary for the chiral transition in*(2 + 1)*-flavor QCD at small values of the chemical potential*,*Phys. Rev.***D 83**(2011) 014504 [arXiv:1011.3130] [INSPIRE]. - [27]tmfT collaboration,
*Thermal QCD transition with two flavors of twisted mass fermions*,*Phys. Rev.***D 87**(2013) 074508 [arXiv:1102.4530] [INSPIRE]. - [28]C.S. Fischer and J.A. Mueller,
*On critical scaling at the QCD N*_{f}= 2*chiral phase transition*,*Phys. Rev.***D 84**(2011) 054013 [arXiv:1106.2700] [INSPIRE]. - [29]C.S. Fischer and J. Luecker,
*Propagators and phase structure of N*_{f}= 2*and N*_{f}= 2 + 1*QCD*,*Phys. Lett.***B 718**(2013) 1036 [arXiv:1206.5191] [INSPIRE]. - [30]M. Grahl, U(2)
_{A}× U(2)_{V}*-symmetric fixed point from the functional renormalization group*,*Phys. Rev.***D 90**(2014) 117904 [arXiv:1410.0985] [INSPIRE]. - [31]Z. Wang and P. Zhuang,
*Critical Behavior and Dimension Crossover of Pion Superfluidity*,*Phys. Rev.***D 94**(2016) 056012 [arXiv:1511.05279] [INSPIRE]. - [32]H.-U. Yee,
*Dynamic universality class of model H with frustrated diffusion: ϵ expansion*,*Phys. Rev.***D 97**(2018) 016003 [arXiv:1707.08560] [INSPIRE]. - [33]M. D’Elia, A. Di Giacomo and C. Pica,
*Two flavor QCD and confinement*,*Phys. Rev.***D 72**(2005) 114510 [hep-lat/0503030] [INSPIRE]. - [34]S. Chandrasekharan and A.C. Mehta,
*Effects of the anomaly on the two-flavor QCD chiral phase transition*,*Phys. Rev. Lett.***99**(2007) 142004 [arXiv:0705.0617] [INSPIRE].ADSCrossRefGoogle Scholar - [35]G. Cossu et al.,
*Finite temperature study of the axial*U(1)*symmetry on the lattice with overlap fermion formulation*,*Phys. Rev.***D 87**(2013) 114514 [*Erratum ibid.***D 88**(2013) 019901] [arXiv:1304.6145] [INSPIRE]. - [36]A. Tomiya, G. Cossu, H. Fukaya, S. Hashimoto and J. Noaki,
*Effects of near-zero Dirac eigenmodes on axial*U(1)*symmetry at finite temperature*, PoS(LATTICE2014)211 (2015) [arXiv:1412.7306] [INSPIRE]. - [37]K. Kanaya and S. Kaya,
*Critical exponents of a three dimensional O*(4)*spin model*,*Phys. Rev.***D 51**(1995) 2404 [hep-lat/9409001] [INSPIRE]. - [38]J. Engels, S. Holtmann, T. Mendes and T. Schulze,
*Finite size scaling functions for 3-D O*(4)*and O*(2)*spin models and QCD*,*Phys. Lett.***B 514**(2001) 299 [hep-lat/0105028] [INSPIRE]. - [39]J. Engels, L. Fromme and M. Seniuch,
*Correlation lengths and scaling functions in the three-dimensional O*(4)*model*,*Nucl. Phys.***B 675**(2003) 533 [hep-lat/0307032] [INSPIRE]. - [40]M. Campostrini, A. Pelissetto, P. Rossi and E. Vicari,
*25th order high temperature expansion results for three-dimensional Ising like systems on the simple cubic lattice*,*Phys. Rev.***E 65**(2002) 066127 [cond-mat/0201180] [INSPIRE]. - [41]A. Bazavov et al.,
*Chiral phase structure of three flavor QCD at vanishing baryon number density*,*Phys. Rev.***D 95**(2017) 074505 [arXiv:1701.03548] [INSPIRE]. - [42]H.-T. Ding,
*Lattice QCD at nonzero temperature and density*, PoS(LATTICE2016)022 (2017) [arXiv:1702.00151] [INSPIRE]. - [43]O. Philipsen,
*Lattice QCD at finite temperature and density*,*Eur. Phys. J. ST***152**(2007) 29 [arXiv:0708.1293] [INSPIRE]. - [44]J.M. Maldacena,
*The Large N limit of superconformal field theories and supergravity*,*Int. J. Theor. Phys.***38**(1999) 1113 [hep-th/9711200] [INSPIRE]. - [45]S.S. Gubser, I.R. Klebanov and A.M. Polyakov,
*Gauge theory correlators from noncritical string theory*,*Phys. Lett.***B 428**(1998) 105 [hep-th/9802109] [INSPIRE]. - [46]E. Witten,
*Anti-de Sitter space and holography*,*Adv. Theor. Math. Phys.***2**(1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [47]O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz,
*Large N field theories, string theory and gravity*,*Phys. Rept.***323**(2000) 183 [hep-th/9905111] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [48]J. Erdmenger, N. Evans, I. Kirsch and E. Threlfall,
*Mesons in Gauge/Gravity Duals — A Review*,*Eur. Phys. J.***A 35**(2008) 81 [arXiv:0711.4467] [INSPIRE]. - [49]G.F. de Teramond and S.J. Brodsky,
*Hadronic Form Factor Models and Spectroscopy Within the Gauge/Gravity Correspondence*, in*Proceedings, Ferrara International School Niccolò Cabeo: Hadron Electromagnetic Form Factors*, Ferrara, Italy, May 23-28, 2011, pp. 54-109 (2011) [arXiv:1203.4025] [INSPIRE]. - [50]A. Adams, L.D. Carr, T. Schäfer, P. Steinberg and J.E. Thomas,
*Strongly Correlated Quantum Fluids: Ultracold Quantum Gases, Quantum Chromodynamic Plasmas and Holographic Duality*,*New J. Phys.***14**(2012) 115009 [arXiv:1205.5180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [51]O. DeWolfe, S.S. Gubser and C. Rosen,
*Dynamic critical phenomena at a holographic critical point*,*Phys. Rev.***D 84**(2011) 126014 [arXiv:1108.2029] [INSPIRE]. - [52]O. DeWolfe, S.S. Gubser and C. Rosen,
*A holographic critical point*,*Phys. Rev.***D 83**(2011) 086005 [arXiv:1012.1864] [INSPIRE]. - [53]S.I. Finazzo, R. Rougemont, M. Zaniboni, R. Critelli and J. Noronha,
*Critical behavior of non-hydrodynamic quasinormal modes in a strongly coupled plasma*,*JHEP***01**(2017) 137 [arXiv:1610.01519] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [54]J. Knaute and B. Kämpfer,
*Holographic Entanglement Entropy in the QCD Phase Diagram with a Critical Point*,*Phys. Rev.***D 96**(2017) 106003 [arXiv:1706.02647] [INSPIRE]. - [55]X. Chen, D. Li and M. Huang,
*Criticality of QCD in a holographic QCD model with critical end point*, arXiv:1810.02136 [INSPIRE]. - [56]I. Iatrakis, E. Kiritsis and A. Paredes,
*An AdS/QCD model from tachyon condensation: II*,*JHEP***11**(2010) 123 [arXiv:1010.1364] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [57]M. Jarvinen and E. Kiritsis,
*Holographic Models for QCD in the Veneziano Limit*,*JHEP***03**(2012) 002 [arXiv:1112.1261] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [58]T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis and K. Tuominen,
*On finite-temperature holographic QCD in the Veneziano limit*,*JHEP***01**(2013) 093 [arXiv:1210.4516] [INSPIRE].ADSCrossRefGoogle Scholar - [59]T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, C. Rosen and K. Tuominen,
*A holographic model for QCD in the Veneziano limit at finite temperature and density*,*JHEP***04**(2014) 124 [*Erratum ibid.***02**(2015) 033] [arXiv:1312.5199] [INSPIRE]. - [60]P. Colangelo, F. Giannuzzi, S. Nicotri and V. Tangorra,
*Temperature and quark density effects on the chiral condensate: An AdS/QCD study*,*Eur. Phys. J.***C 72**(2012) 2096 [arXiv:1112.4402] [INSPIRE]. - [61]N. Evans, C. Miller and M. Scott,
*Inverse Magnetic Catalysis in Bottom-Up Holographic QCD*,*Phys. Rev.***D 94**(2016) 074034 [arXiv:1604.06307] [INSPIRE]. - [62]D. Dudal, D.R. Granado and T.G. Mertens,
*No inverse magnetic catalysis in the QCD hard and soft wall models*,*Phys. Rev.***D 93**(2016) 125004 [arXiv:1511.04042] [INSPIRE]. - [63]K. Chelabi, Z. Fang, M. Huang, D. Li and Y.-L. Wu,
*Realization of chiral symmetry breaking and restoration in holographic QCD*,*Phys. Rev.***D 93**(2016) 101901 [arXiv:1511.02721] [INSPIRE]. - [64]K. Chelabi, Z. Fang, M. Huang, D. Li and Y.-L. Wu,
*Chiral Phase Transition in the Soft-Wall Model of AdS/QCD*,*JHEP***04**(2016) 036 [arXiv:1512.06493] [INSPIRE].ADSGoogle Scholar - [65]Z. Fang, S. He and D. Li,
*Chiral and Deconfining Phase Transitions from Holographic QCD Study*,*Nucl. Phys.***B 907**(2016) 187 [arXiv:1512.04062] [INSPIRE]. - [66]D. Li, M. Huang, Y. Yang and P.-H. Yuan,
*Inverse Magnetic Catalysis in the Soft-Wall Model of AdS/QCD*,*JHEP***02**(2017) 030 [arXiv:1610.04618] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [67]D. Li and M. Huang,
*Chiral phase transition of QCD with N*_{f}= 2 + 1*flavors from holography*,*JHEP***02**(2017) 042 [arXiv:1610.09814] [INSPIRE]. - [68]S.P. Bartz and T. Jacobson,
*Chiral Phase Transition and Meson Melting from AdS/QCD*,*Phys. Rev.***D 94**(2016) 075022 [arXiv:1607.05751] [INSPIRE]. - [69]Z. Fang, Y.-L. Wu and L. Zhang,
*Chiral phase transition and meson spectrum in improved soft-wall AdS/QCD*,*Phys. Lett.***B 762**(2016) 86 [arXiv:1604.02571] [INSPIRE]. - [70]S.P. Bartz and T. Jacobson,
*Chiral phase transition at finite chemical potential in 2+1 -flavor soft-wall anti-de Sitter space QCD*,*Phys. Rev.***C 97**(2018) 044908 [arXiv:1801.00358] [INSPIRE]. - [71]Z. Fang, Y.-L. Wu and L. Zhang,
*Chiral Phase Transition with 2+1 quark flavors in an improved soft-wall AdS/QCD Model*,*Phys. Rev.***D 98**(2018) 114003 [arXiv:1805.05019] [INSPIRE]. - [72]U. Gürsoy, I. Iatrakis, M. Järvinen and G. Nijs,
*Inverse Magnetic Catalysis from improved Holographic QCD in the Veneziano limit*,*JHEP***03**(2017) 053 [arXiv:1611.06339] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [73]A. Karch, E. Katz, D.T. Son and M.A. Stephanov,
*Linear confinement and AdS/QCD*,*Phys. Rev.***D 74**(2006) 015005 [hep-ph/0602229] [INSPIRE]. - [74]N. Evans, K. Jensen and K.-Y. Kim,
*Non Mean-Field Quantum Critical Points from Holography*,*Phys. Rev.***D 82**(2010) 105012 [arXiv:1008.1889] [INSPIRE]. - [75]T. Gherghetta, J.I. Kapusta and T.M. Kelley,
*Chiral symmetry breaking in the soft-wall AdS/QCD model*,*Phys. Rev.***D 79**(2009) 076003 [arXiv:0902.1998] [INSPIRE]. - [76]T.M. Kelley, S.P. Bartz and J.I. Kapusta,
*Pseudoscalar Mass Spectrum in a Soft-Wall Model of AdS/QCD*,*Phys. Rev.***D 83**(2011) 016002 [arXiv:1009.3009] [INSPIRE]. - [77]Y.-Q. Sui, Y.-L. Wu, Z.-F. Xie and Y.-B. Yang,
*Prediction for the Mass Spectra of Resonance Mesons in the Soft-Wall AdS/QCD with a Modified 5D Metric*,*Phys. Rev.***D 81**(2010) 014024 [arXiv:0909.3887] [INSPIRE]. - [78]Y.-Q. Sui, Y.-L. Wu and Y.-B. Yang,
*Predictive AdS/QCD Model for Mass Spectra of Mesons with Three Flavors*,*Phys. Rev.***D 83**(2011) 065030 [arXiv:1012.3518] [INSPIRE]. - [79]L.-X. Cui, Z. Fang and Y.-L. Wu,
*Infrared-Improved Soft-wall AdS/QCD Model for Mesons*,*Eur. Phys. J.***C 76**(2016) 22 [arXiv:1310.6487] [INSPIRE]. - [80]D. Li, M. Huang and Q.-S. Yan,
*A dynamical soft-wall holographic QCD model for chiral symmetry breaking and linear confinement*,*Eur. Phys. J.***C 73**(2013) 2615 [arXiv:1206.2824] [INSPIRE]. - [81]D. Li, M. Huang and Q.-S. Yan,
*Accommodate chiral symmetry breaking and linear confinement in a dynamical holographic QCD model*,*AIP Conf. Proc.***1492**(2012) 233 [arXiv:1209.1202] [INSPIRE]. - [82]D. Li and M. Huang,
*Dynamical holographic QCD model for glueball and light meson spectra*,*JHEP***11**(2013) 088 [arXiv:1303.6929] [INSPIRE].ADSCrossRefGoogle Scholar - [83]A. Vega and P. Cabrera,
*Family of dilatons and metrics for AdS/QCD models*,*Phys. Rev.***D 93**(2016) 114026 [arXiv:1601.05999] [INSPIRE]. - [84]E. Folco Capossoli and H. Boschi-Filho,
*Glueball spectra and Regge trajectories from a modified holographic softwall model*,*Phys. Lett.***B 753**(2016) 419 [arXiv:1510.03372] [INSPIRE]. - [85]E. Folco Capossoli, D. Li and H. Boschi-Filho,
*Pomeron and Odderon Regge Trajectories from a Dynamical Holographic Model*,*Phys. Lett.***B 760**(2016) 101 [arXiv:1601.05114] [INSPIRE]. - [86]E. Folco Capossoli, D. Li and H. Boschi-Filho,
*Dynamical corrections to the anomalous holographic soft-wall model: the Pomeron and the odderon*,*Eur. Phys. J.***C 76**(2016) 320 [arXiv:1604.01647] [INSPIRE]. - [87]R. Zöllner and B. Kampfer,
*Extended soft wall model with background related to features of QCD thermodynamics*,*Eur. Phys. J.***A 53**(2017) 139 [arXiv:1701.01398] [INSPIRE]. - [88]S. He, S.-Y. Wu, Y. Yang and P.-H. Yuan,
*Phase Structure in a Dynamical Soft-Wall Holographic QCD Model*,*JHEP***04**(2013) 093 [arXiv:1301.0385] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [89]D. Li, S. He, M. Huang and Q.-S. Yan,
*Thermodynamics of deformed AdS*_{5}*model with a positive/negative quadratic correction in graviton-dilaton system*,*JHEP***09**(2011) 041 [arXiv:1103.5389] [INSPIRE]. - [90]R.-G. Cai, S. He and D. Li,
*A hQCD model and its phase diagram in Einstein-Maxwell-Dilaton system*,*JHEP***03**(2012) 033 [arXiv:1201.0820] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [91]A. Cherman, T.D. Cohen and E.S. Werbos,
*The Chiral condensate in holographic models of QCD*,*Phys. Rev.***C 79**(2009) 045203 [arXiv:0804.1096] [INSPIRE]. - [92]S.A. Hartnoll, C.P. Herzog and G.T. Horowitz,
*Building a Holographic Superconductor*,*Phys. Rev. Lett.***101**(2008) 031601 [arXiv:0803.3295] [INSPIRE]. - [93]U. Gürsoy,
*Continuous Hawking-Page transitions in Einstein-scalar gravity*,*JHEP***01**(2011) 086 [arXiv:1007.0500] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar