Journal of High Energy Physics

, 2013:142 | Cite as

Entanglement temperature in non-conformal cases

  • Song He
  • Danning Li
  • Jun-Bao Wu


Potential reconstruction can be used to find various analytical asymptotical AdS solutions in Einstein dilation system generally. We have generated two simple solutions without physical singularity called zero temperature solutions. We also proposed a numerical way to obtain black hole solution in Einstein dilaton system with special dilaton potential. By using this method, we obtain the corresponding black hole solutions numerically and investigate the thermal stability of the black hole by comparing the free energy of thermal gas and the corresponding black hole. In two groups of non-conformal gravity solutions obtained in this paper, we find that the two thermal gas solutions are more unstable than black hole solutions respectively. Finally, we consider black hole solutions as a thermal state of zero temperature solutions to check that the first thermal dynamical law exists in entanglement system from holographic point of view.


Gauge-gravity correspondence AdS-CFT Correspondence 


  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  2. [2]
    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].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  4. [4]
    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].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  9. [9]
    T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
  10. [10]
    T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
  11. [11]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetGoogle Scholar
  12. [12]
    T. Takayanagi, Entanglement entropy from a holographic viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    T. Albash and C.V. Johnson, Holographic entanglement entropy and renormalization group flow, JHEP 02 (2012) 095 [arXiv:1110.1074] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    R.C. Myers and A. Singh, Comments on holographic entanglement entropy and RG flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    J. de Boer, M. Kulaxizi and A. Parnachev, Holographic entanglement entropy in Lovelock gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    L.-Y. Hung, R.C. Myers and M. Smolkin, On holographic entanglement entropy and higher curvature gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    B. Chen and J.-j. Zhang, Note on generalized gravitational entropy in Lovelock gravity, JHEP 07 (2013) 185 [arXiv:1305.6767] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A. Bhattacharyya, A. Kaviraj and A. Sinha, Entanglement entropy in higher derivative holography, JHEP 08 (2013) 012 [arXiv:1305.6694] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    T. Nishioka and T. Takayanagi, AdS bubbles, entropy and closed string tachyons, JHEP 01 (2007) 090 [hep-th/0611035] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    J.-R. Sun, Note on Chern-Simons term correction to holographic entanglement entropy, JHEP 05 (2009) 061 [arXiv:0810.0967] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    A. Pakman and A. Parnachev, Topological entanglement entropy and holography, JHEP 07 (2008) 097 [arXiv:0805.1891] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    N. Ogawa and T. Takayanagi, Higher derivative corrections to holographic entanglement entropy for AdS solitons, JHEP 10 (2011) 147 [arXiv:1107.4363] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    R.-G. Cai, S. He, L. Li and Y.-L. Zhang, Holographic entanglement entropy in insulator/superconductor transition, JHEP 07 (2012) 088 [arXiv:1203.6620] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    R.-G. Cai, S. He, L. Li and Y.-L. Zhang, Holographic entanglement entropy on P-wave superconductor phase transition, JHEP 07 (2012) 027 [arXiv:1204.5962] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    R.-G. Cai, S. He, L. Li and L.-F. Li, Entanglement entropy and Wilson loop in Stúckelberg holographic insulator/superconductor model, JHEP 10 (2012) 107 [arXiv:1209.1019] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    M. Nozaki, T. Numasawa and T. Takayanagi, Holographic local quenches and entanglement density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    M. Nozaki, T. Numasawa, A. Prudenziati and T. Takayanagi, Dynamics of Entanglement Entropy from Einstein Equation, Phys. Rev. D 88 (2013) 026012 [arXiv:1304.7100] [INSPIRE].ADSGoogle Scholar
  30. [30]
    M.B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. 8 (2007) 24 [arXiv:0705.2024].Google Scholar
  31. [31]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  32. [32]
    F.C. Alcaraz, M.I. Berganza and G. Sierra, Entanglement of low-energy excitations in Conformal Field Theory, Phys. Rev. Lett. 106 (2011) 201601 [arXiv:1101.2881] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    L. Masanes, An area law for the entropy of low-energy states, Phys. Rev. A 80 (2009) 052104 [arXiv:0907.4672] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    D. Allahbakhshi, M. Alishahiha and A. Naseh, Entanglement thermodynamics, JHEP 08 (2013) 102 [arXiv:1305.2728] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical property of entanglement entropy for excited states, Phys. Rev. Lett. 110 (2013), no. 9 091602 [arXiv:1212.1164] [INSPIRE].
  36. [36]
    R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].ADSGoogle Scholar
  37. [37]
    W.-z. Guo, S. He and J. Tao, Note on entanglement temperature for low thermal excited states in higher derivative gravity, JHEP 08 (2013) 050 [arXiv:1305.2682] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].Google Scholar
  40. [40]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, arXiv:1307.2892 [INSPIRE].
  41. [41]
    S.S. Gubser and A. Nellore, Mimicking the QCD equation of state with a dual black hole, Phys. Rev. D 78 (2008) 086007 [arXiv:0804.0434] [INSPIRE].ADSGoogle Scholar
  42. [42]
    S.S. Gubser, A. Nellore, S.S. Pufu and F.D. Rocha, Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics, Phys. Rev. Lett. 101 (2008) 131601 [arXiv:0804.1950] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    S.S. Gubser, S.S. Pufu and F.D. Rocha, Bulk viscosity of strongly coupled plasmas with holographic duals, JHEP 08 (2008) 085 [arXiv:0806.0407] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Deconfinement and Gluon Plasma Dynamics in Improved Holographic QCD, Phys. Rev. Lett. 101 (2008) 181601 [arXiv:0804.0899] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    U. Gürsoy, E. Kiritsis, G. Michalogiorgakis and F. Nitti, Thermal transport and drag force in improved holographic QCD, JHEP 12 (2009) 056 [arXiv:0906.1890] [INSPIRE].CrossRefGoogle Scholar
  46. [46]
    K. Farakos, A. Kouretsis and P. Pasipoularides, Anti de Sitter 5D black hole solutions with a self-interacting bulk scalar field: a potential reconstruction approach, Phys. Rev. D 80 (2009) 064020 [arXiv:0905.1345] [INSPIRE].ADSGoogle Scholar
  47. [47]
    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].ADSCrossRefGoogle Scholar
  48. [48]
    N. Ohta and T. Torii, Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions IV: Topological Black Holes with and without Cosmological Term, Prog. Theor. Phys. 122 (2009) 1477 [arXiv:0908.3918] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, A New Class of Exact Hairy Black Hole Solutions, Gen. Rel. Grav. 43 (2011) 163 [arXiv:0911.1711] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  50. [50]
    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].ADSCrossRefGoogle Scholar
  51. [51]
    S. He, Y.-P. Hu and J.-H. Zhang, Hydrodynamics of a 5D Einstein-dilaton black hole solution and the corresponding BPS state, JHEP 12 (2011) 078 [arXiv:1111.1374] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    S. He, M. Huang and Q.-S. Yan, Logarithmic correction in the deformed AdS 5 model to produce the heavy quark potential and QCD β-function, Phys. Rev. D 83 (2011) 045034 [arXiv:1004.1880] [INSPIRE].ADSGoogle Scholar
  53. [53]
    R.-G. Cai, S. Chakrabortty, S. He and L. Li, Some aspects of QGP phase in a hQCD model, JHEP 02 (2013) 068 [arXiv:1209.4512] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  56. [56]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    K. Narayan, Non-conformal brane plane waves and entanglement entropy, Phys. Lett. B 726 (2013) 370 [arXiv:1304.6697] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    U. Gürsoy, E. Kiritsis, L. Mazzanti and F. Nitti, Holography and thermodynamics of 5D dilaton-gravity, JHEP 05 (2009) 033 [arXiv:0812.0792] [INSPIRE].CrossRefGoogle Scholar
  59. [59]
    P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.State Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of ScienceBeijingP.R. China
  2. 2.Institute of High Energy Physics, and Theoretical Physics Center for Science FacilitiesChinese Academy of SciencesBeijingP.R. China

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