Bulk viscosity in holographic Lifshitz hydrodynamics

  • Carlos Hoyos
  • Bom Soo Kim
  • Yaron Oz
Open Access


We compute the bulk viscosity in holographic models dual to theories with Lifshitz scaling and/or hyperscaling violation, using a generalization of the bulk viscosity formula derived in arXiv:1103.1657 from the null focusing equation. We find that only a class of models with massive vector fields are truly Lifshitz scale invariant, and have a vanishing bulk viscosity. For other holographic models with scalars and/or massless vector fields we find a universal formula in terms of the dynamical exponent and the hyperscaling violation exponent.


Gauge-gravity correspondence Space-Time Symmetries Holography and condensed matter physics (AdS/CMT) 


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.


  1. [1]
    R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    C. Hoyos, B.S. Kim and Y. Oz, Lifshitz hydrodynamics, JHEP 11 (2013) 145 [arXiv:1304.7481] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    C. Hoyos, B.S. Kim and Y. Oz, Lifshitz field theories at non-zero temperature, hydrodynamics and gravity, arXiv:1309.6794 [INSPIRE].
  4. [4]
    I. Kanitscheider, K. Skenderis and M. Taylor, Precision holography for non-conformal branes, JHEP 09 (2008) 094 [arXiv:0807.3324] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    I. Kanitscheider and K. Skenderis, Universal hydrodynamics of non-conformal branes, JHEP 04 (2009) 062 [arXiv:0901.1487] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    B. Gouteraux and E. Kiritsis, Generalized holographic quantum criticality at finite density, JHEP 12 (2011) 036 [arXiv:1107.2116] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
  9. [9]
    J. Gath, J. Hartong, R. Monteiro and N.A. Obers, Holographic models for theories with hyperscaling violation, JHEP 04 (2013) 159 [arXiv:1212.3263] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    C. Eling and Y. Oz, A novel formula for bulk viscosity from the null horizon focusing equation, JHEP 06 (2011) 007 [arXiv:1103.1657] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    C. Eling and Y. Oz, Holographic screens and transport coefficients in the fluid/gravity correspondence, Phys. Rev. Lett. 107 (2011) 201602 [arXiv:1107.2134] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    C. Eling, I. Fouxon and Y. Oz, The incompressible Navier-Stokes equations from membrane dynamics, Phys. Lett. B 680 (2009) 496 [arXiv:0905.3638] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    C. Eling and Y. Oz, Relativistic CFT hydrodynamics from the membrane paradigm, JHEP 02 (2010) 069 [arXiv:0906.4999] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    C. Eling, Y. Neiman and Y. Oz, Holographic non-abelian charged hydrodynamics from the dynamics of null horizons, JHEP 12 (2010) 086 [arXiv:1010.1290] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    J. Mas and J. Tarrio, Hydrodynamics from the Dp-brane, JHEP 05 (2007) 036 [hep-th/0703093] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    P. Benincasa and A. Buchel, Hydrodynamics of Sakai-Sugimoto model in the quenched approximation, Phys. Lett. B 640 (2006) 108 [hep-th/0605076] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. Benincasa, A. Buchel and A.O. Starinets, Sound waves in strongly coupled non-conformal gauge theory plasma, Nucl. Phys. B 733 (2006) 160 [hep-th/0507026] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    A. Buchel and C. Pagnutti, Bulk viscosity of N = 2 plasma, Nucl. Phys. B 816 (2009) 62 [arXiv:0812.3623] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    A. Buchel, Critical phenomena in N = 4 SYM plasma, Nucl. Phys. B 841 (2010) 59 [arXiv:1005.0819] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    A. Yarom, Notes on the bulk viscosity of holographic gauge theory plasmas, JHEP 04 (2010) 024 [arXiv:0912.2100] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    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
  22. [22]
    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
  23. [23]
    K. Balasubramanian and J. McGreevy, An analytic Lifshitz black hole, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    Y. Korovin, K. Skenderis and M. Taylor, Lifshitz from AdS at finite temperature and top down models, JHEP 11 (2013) 127 [arXiv:1306.3344] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    G. Bertoldi, B.A. Burrington and A. Peet, Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    G. Bertoldi, B.A. Burrington and A.W. Peet, Thermodynamics of black branes in asymptotically Lifshitz spacetimes, Phys. Rev. D 80 (2009) 126004 [arXiv:0907.4755] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    C. Charmousis, B. Gouteraux and J. Soda, Einstein-Maxwell-dilaton theories with a Liouville potential, Phys. Rev. D 80 (2009) 024028 [arXiv:0905.3337] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    D.-W. Pang, A note on black holes in asymptotically Lifshitz spacetime, arXiv:0905.2678 [INSPIRE].
  30. [30]
    S.S. Gubser and F.D. Rocha, Peculiar properties of a charged dilatonic black hole in AdS 5, Phys. Rev. D 81 (2010) 046001 [arXiv:0911.2898] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    K. Goldstein, S. Kachru, S. Prakash and S.P. Trivedi, Holography of charged dilaton black holes, JHEP 08 (2010) 078 [arXiv:0911.3586] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    K. Goldstein et al., Holography of dyonic dilaton black branes, JHEP 10 (2010) 027 [arXiv:1007.2490] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    C. Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis and R. Meyer, Effective holographic theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    N. Iizuka, N. Kundu, P. Narayan and S.P. Trivedi, Holographic Fermi and non-Fermi liquids with transitions in dilaton gravity, JHEP 01 (2012) 094 [arXiv:1105.1162] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Cadoni, S. Mignemi and M. Serra, Exact solutions with AdS asymptotics of Einstein and Einstein-Maxwell gravity minimally coupled to a scalar field, Phys. Rev. D 84 (2011) 084046 [arXiv:1107.5979] [INSPIRE].ADSGoogle Scholar
  36. [36]
    J. Tarrio and S. Vandoren, Black holes and black branes in Lifshitz spacetimes, JHEP 09 (2011) 017 [arXiv:1105.6335] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    M. Alishahiha, E. O’Colgain and H. Yavartanoo, Charged black branes with hyperscaling violating factor, JHEP 11 (2012) 137 [arXiv:1209.3946] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A. Buchel, Bulk viscosity of gauge theory plasma at strong coupling, Phys. Lett. B 663 (2008) 286 [arXiv:0708.3459] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    R.B. Mann, Lifshitz topological black holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    E. Brynjolfsson, U. Danielsson, L. Thorlacius and T. Zingg, Holographic superconductors with Lifshitz scaling, J. Phys. A 43 (2010) 065401 [arXiv:0908.2611] [INSPIRE].ADSMathSciNetGoogle Scholar
  42. [42]
    D.-W. Pang, On charged Lifshitz black holes, JHEP 01 (2010) 116 [arXiv:0911.2777] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M.H. Dehghani, R.B. Mann and R. Pourhasan, Charged Lifshitz black holes, Phys. Rev. D 84 (2011) 046002 [arXiv:1102.0578] [INSPIRE].ADSGoogle Scholar
  44. [44]
    L. Barclay, R. Gregory, S. Parameswaran, G. Tasinato and I. Zavala, Lifshitz black holes in IIA supergravity, JHEP 05 (2012) 122 [arXiv:1203.0576] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    B. Gouteraux and E. Kiritsis, Quantum critical lines in holographic phases with (un)broken symmetry, JHEP 04 (2013) 053 [arXiv:1212.2625] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    B. Gouteraux, J. Smolic, M. Smolic, K. Skenderis and M. Taylor, Holography for Einstein-Maxwell-dilaton theories from generalized dimensional reduction, JHEP 01 (2012) 089 [arXiv:1110.2320] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  47. [47]
    M. Cadoni, G. D’Appollonio and P. Pani, Phase transitions between Reissner-Nordstrom and dilatonic black holes in 4D AdS spacetime, JHEP 03 (2010) 100 [arXiv:0912.3520] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    G. Bertoldi, B.A. Burrington and A.W. Peet, Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries, Phys. Rev. D 82 (2010) 106013 [arXiv:1007.1464] [INSPIRE].ADSGoogle Scholar
  49. [49]
    G. Bertoldi, B.A. Burrington, A.W. Peet and I.G. Zadeh, Lifshitz-like black brane thermodynamics in higher dimensions, Phys. Rev. D 83 (2011) 126006 [arXiv:1101.1980] [INSPIRE].ADSGoogle Scholar
  50. [50]
    W. Chemissany and J. Hartong, From D3-branes to Lifshitz Space-Times, Class. Quant. Grav. 28 (2011) 195011 [arXiv:1105.0612] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    P. Berglund, J. Bhattacharyya and D. Mattingly, Charged dilatonic AdS black branes in arbitrary dimensions, JHEP 08 (2012) 042 [arXiv:1107.3096] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi surfaces and entanglement entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  53. [53]
    L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    T. Azeyanagi, W. Li and T. Takayanagi, On string theory duals of Lifshitz-like fixed points, JHEP 06 (2009) 084 [arXiv:0905.0688] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    S. Sachdev, Quantum phase transitions, 2nd edition, Cambridge University Press, Cambridge U.K. (2011).CrossRefzbMATHGoogle Scholar
  56. [56]
    P. Gegenwart, Q. Si and F. Steglich, Quantum criticality in heavy-fermion metals, Nature Phys. 4 (2008) 186 [arXiv:0712.2045].ADSCrossRefGoogle Scholar
  57. [57]
    R. Hornreich, M. Luban and S. Shtrikman, Critical behavior at the onset of \( \overrightarrow{k} \) -space instability on the λ line, Phys. Rev. Lett. 35 (1975) 1678 [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Raymond and Beverly Sackler School of Physics and AstronomyTel-Aviv UniversityTel-AvivIsrael

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