Journal of High Energy Physics

, 2018:87 | Cite as

Semi-local quantum criticality and the instability of extremal planar horizons

  • Samuel E. Gralla
  • Arun Ravishankar
  • Peter ZimmermanEmail author
Open Access
Regular Article - Theoretical Physics


We show that the Aretakis instability of compact extremal horizons persists in the planar case of interest to holography and discuss its connection with the emergence of “semi-local quantum criticality” in the field theory dual. In particular, the spatially localized power-law decay of this critical phase corresponds to spatially localized power-law growth of stress-energy on the horizon. For near-extremal black holes these phenomena occur transiently over times of order the inverse temperature. The boundary critical phase is characterized by an emergent temporal conformal symmetry, and the bulk instability seems to be essential to preserving the symmetry in the presence of interactions. We work primarily in the solvable example of charged scalar perturbations of five-dimensional (near-)extremal planar Reissner-Nordström anti-de Sitter spacetime and argue that the conclusions hold more generally.


AdS-CFT Correspondence Black Holes 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]
    S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
  2. [2]
    G.R. Stewart, Non-Fermi-liquid behavior in d- and f-electron metals, Rev. Mod. Phys. 73 (2001) 797 [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, Strange metal transport realized by gauge/gravity duality, Science 329 (2010) 1043 [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    N. Iqbal, H. Liu and M. Mezei, Lectures on holographic non-Fermi liquids and quantum phase transitions, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2010). String Theory and Its Applications: From meV to the Planck Scale, Boulder U.S.A. (2010), pg. 707 [arXiv:1110.3814] [INSPIRE].
  5. [5]
    S. Aretakis, The Wave Equation on Extreme Reissner-Nordstrom Black Hole Spacetimes: Stability and Instability Results, arXiv:1006.0283 [INSPIRE].
  6. [6]
    S. Aretakis, Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I, Commun. Math. Phys. 307 (2011) 17 [arXiv:1110.2007] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Aretakis, Stability and Instability of Extreme Reissner-Nordstrom Black Hole Spacetimes for Linear Scalar Perturbations II, Annales Henri Poincaré 12 (2011) 1491 [arXiv:1110.2009] [INSPIRE].
  8. [8]
    J. Lucietti, K. Murata, H.S. Reall and N. Tanahashi, On the horizon instability of an extreme Reissner-Nordström black hole, JHEP 03 (2013) 035 [arXiv:1212.2557] [INSPIRE].CrossRefzbMATHGoogle Scholar
  9. [9]
    S. Aretakis, A note on instabilities of extremal black holes under scalar perturbations from afar, Class. Quant. Grav. 30 (2013) 095010 [arXiv:1212.1103] [INSPIRE].
  10. [10]
    M. Casals, S.E. Gralla and P. Zimmerman, Horizon Instability of Extremal Kerr Black Holes: Nonaxisymmetric Modes and Enhanced Growth Rate, Phys. Rev. D 94 (2016) 064003 [arXiv:1606.08505] [INSPIRE].
  11. [11]
    Y. Angelopoulos, S. Aretakis and D. Gajic, Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds, arXiv:1807.03802 [INSPIRE].
  12. [12]
    K. Murata, H.S. Reall and N. Tanahashi, What happens at the horizon(s) of an extreme black hole?, Class. Quant. Grav. 30 (2013) 235007 [arXiv:1307.6800] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S.E. Gralla, A. Zimmerman and P. Zimmerman, Transient Instability of Rapidly Rotating Black Holes, Phys. Rev. D 94 (2016) 084017 [arXiv:1608.04739] [INSPIRE].
  14. [14]
    P. Zimmerman, Horizon instability of extremal Reissner-Nordström black holes to charged perturbations, Phys. Rev. D 95 (2017) 124032 [arXiv:1612.03172] [INSPIRE].
  15. [15]
    S. Aretakis, Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds, J. Funct. Anal. 263 (2012) 2770 [arXiv:1110.2006] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    S. Aretakis, Horizon Instability of Extremal Black Holes, Adv. Theor. Math. Phys. 19 (2015) 507 [arXiv:1206.6598] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Lucietti and H.S. Reall, Gravitational instability of an extreme Kerr black hole, Phys. Rev. D 86 (2012) 104030 [arXiv:1208.1437] [INSPIRE].
  18. [18]
    S.E. Gralla and P. Zimmerman, Scaling and Universality in Extremal Black Hole Perturbations, JHEP 06 (2018) 061 [arXiv:1804.04753] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruckenstein, Phenomenology of the normal state of Cu-O high-temperature superconductors, Phys. Rev. Lett. 63 (1989) 1996 [INSPIRE].CrossRefGoogle Scholar
  20. [20]
    A. Schröder et al., Onset of antiferromagnetism in heavy-fermion metals, Nature 407 (2000) 351.CrossRefGoogle Scholar
  21. [21]
    Q. Si, S. Rabello, K. Ingersent and J.L. Smith, Locally critical quantum phase transitions in strongly correlated metals, Nature 413 (2001) 804.CrossRefGoogle Scholar
  22. [22]
    N. Iqbal, H. Liu and M. Mezei, Semi-local quantum liquids, JHEP 04 (2012) 086 [arXiv:1105.4621] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    S.E. Gralla and P. Zimmerman, Critical Exponents of Extremal Kerr Perturbations, Class. Quant. Grav. 35 (2018) 095002 [arXiv:1711.00855] [INSPIRE].
  24. [24]
    S. Hadar and H.S. Reall, Is there a breakdown of effective field theory at the horizon of an extremal black hole?, JHEP 12 (2017) 062 [arXiv:1709.09668] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    L.M. Burko and G. Khanna, Linearized Stability of Extreme Black Holes, Phys. Rev. D 97 (2018) 061502 [arXiv:1709.10155] [INSPIRE].
  26. [26]
    A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018 [hep-th/9902170] [INSPIRE].
  27. [27]
    T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].
  28. [28]
    D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  29. [29]
    J. Ren, Analytic quantum critical points from holography, arXiv:1210.2722 [INSPIRE].
  30. [30]
    J. Ren, Quantum Critical Systems from AdS/CFT, Ph.D. Thesis, Princeton University, Princeton U.S.A. (2013).Google Scholar
  31. [31]
    DLMF, NIST Digital Library of Mathematical Functions,, Release 1.0.5 of 2012-10-01.
  32. [32]
    N. Iqbal, H. Liu, M. Mezei and Q. Si, Quantum phase transitions in holographic models of magnetism and superconductors, Phys. Rev. D 82 (2010) 045002 [arXiv:1003.0010] [INSPIRE].
  33. [33]
    F. Denef and S.A. Hartnoll, Landscape of superconducting membranes, Phys. Rev. D 79 (2009) 126008 [arXiv:0901.1160] [INSPIRE].
  34. [34]
    S.E. Gralla, S.A. Hughes and N. Warburton, Inspiral into Gargantua, Class. Quant. Grav. 33 (2016) 155002 [arXiv:1603.01221] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    G. Compère, K. Fransen, T. Hertog and J. Long, Gravitational waves from plunges into Gargantua, Class. Quant. Grav. 35 (2018) 104002 [arXiv:1712.07130] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    B.M. Project, H. Bateman, A. Erdélyi and U.S.O. of Naval Research, Tables of Integral Transforms: Based, in Part, on Notes Left by Harry Bateman. Vol. 2, McGraw-Hill, New York U.S.A. (1954).Google Scholar
  37. [37]
    T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, Holographic non-Fermi-liquid fixed points, Phil. Trans. Roy. Soc. A 369 (2011) 1640 [arXiv:1101.0597].
  38. [38]
    S. Hod, Slow relaxation of rapidly rotating black holes, Phys. Rev. D 78 (2008) 084035 [arXiv:0811.3806] [INSPIRE].
  39. [39]
    H. Yang, A. Zimmerman, A. Zenginoğlu, F. Zhang, E. Berti and Y. Chen, Quasinormal modes of nearly extremal Kerr spacetimes: spectrum bifurcation and power-law ringdown, Phys. Rev. D 88 (2013) 044047 [arXiv:1307.8086] [INSPIRE].
  40. [40]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  41. [41]
    D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
  42. [42]
    F.W. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions, first edition, Cambridge University Press, New York U.S.A. (2010),

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Samuel E. Gralla
    • 1
  • Arun Ravishankar
    • 1
  • Peter Zimmerman
    • 1
    Email author
  1. 1.Department of PhysicsUniversity of ArizonaTucsonU.S.A.

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