A thermal quench induces spatial inhomogeneities in a holographic superconductor

  • Antonio M. García-García
  • Hua Bi Zeng
  • Hai-Qing Zhang
Open Access


Holographic duality is a powerful tool to investigate the far-from equilibrium dynamics of superfluids and other phases of quantum matter. For technical reasons it is usually assumed that, after a quench, the far-from equilibrium fields are still spatially uniform. Here we relax this assumption and study the time evolution of a holographic superconductor after a temperature quench but allowing spatial variations of the order parameter. Even though the initial state and the quench are spatially uniform we show the order parameter develops spatial oscillations with an amplitude that increases with time until it reaches a stationary value. The free energy of these inhomogeneous solutions is lower than that of the homogeneous ones. Therefore the former corresponds to the physical configuration that could be observed experimentally.


Duality in Gauge Field Theories AdS-CFT Correspondence Black Holes 


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]
    C.L. Smallwood et al., Tracking Cooper Pairs in a Cuprate Superconductor by Ultrafast Angle-Resolved Photoemission, Science 336 (2012) 1137 [arXiv:1206.2300].ADSCrossRefGoogle Scholar
  2. [2]
    T. Rohwer et al., Collapse of long-range charge order tracked by time-resolved photoemission at high momenta, Nature 471 (2011) 490.ADSCrossRefGoogle Scholar
  3. [3]
    D. Fausti et al., Light-Induced Superconductivity in a Stripe-Ordered Cuprate, Science 331 (2011) 189.ADSCrossRefGoogle Scholar
  4. [4]
    C.N. Weiler et al., Spontaneous vortices in the formation of Bose-Einstein condensates, Nature 455 (2008) 948 [arXiv:0807.3323].ADSCrossRefGoogle Scholar
  5. [5]
    T.W.B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387 [INSPIRE].ADSGoogle Scholar
  6. [6]
    W.H. Zurek, Cosmological Experiments in Superfluid Helium?, Nature 317 (1985) 505 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    P. Laguna and W.H. Zurek, Density of kinks after a quench: When symmetry breaks, how big are the pieces?, Phys. Rev. Lett. 78 (1997) 2519 [gr-qc/9607041] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    R.A. Barankov and L.S. Levitov, Synchronization in the BCS Pairing Dynamics as a Critical Phenomenon, Phys. Rev. Lett. 96 (2006) 230403 [cond-mat/0603317].ADSCrossRefGoogle Scholar
  9. [9]
    E.A. Yuzbashyan and M. Dzero, Dynamical Vanishing of the Order Parameter in a Fermionic Condensate, Phys. Rev. Lett. 96 (2006) 230404 [cond-mat/0603404].ADSCrossRefGoogle Scholar
  10. [10]
    R.A. Barankov, L.S. Levitov and B.Z. Spivak, Collective Rabi Oscillations and Solitons in a Time-Dependent BCS Pairing Problem, Phys. Rev. Lett. 93 (2004) 160401 [cond-mat/0312053].ADSCrossRefGoogle Scholar
  11. [11]
    M. Dzero, E.A. Yuzbashyan and B.L. Altshuler, Cooper pair turbulence in atomic Fermi gases, Eur. Phys. Lett. 85 (2009) 20004. [arXiv:0805.2798].ADSCrossRefGoogle Scholar
  12. [12]
    J. Dziarmaga, Dynamics of a Quantum Phase Transition: Exact Solution of the Quantum Ising Model, Phys. Rev. Lett. 95 (2005) 245701.ADSCrossRefGoogle Scholar
  13. [13]
    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].
  14. [14]
    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].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMATHMathSciNetGoogle Scholar
  16. [16]
    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].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    P.M. Chesler and L.G. Yaffe, Horizon Formation and Far-from-Equilibrium Isotropization in a Supersymmetric Yang-Mills Plasma, Phys. Rev. Lett. 102 211601 (2009) [arXiv:0812.2053] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 191601 (2011) [arXiv:1012.4753] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    P. Bizon and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys. Rev. Lett. 107 (2011) 031102 [arXiv:1104.3702] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    O.J.C. Dias, G.T. Horowitz, D. Marolf and J.E. Santos, On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions, Class. Quant. Grav. 29 (2012) 235019 [arXiv:1208.5772] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    K. Murata, S. Kinoshita and N. Tanahashi, Non-equilibrium Condensation Process in a Holographic Superconductor, JHEP 07 (2010) 050 [arXiv:1005.0633] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M.J. Bhaseen, J.P. Gauntlett, B.D. Simons, J. Sonner and T. Wiseman, Holographic Superfluids and the Dynamics of Symmetry Breaking, Phys. Rev. Lett. 110 (2013) 015301 [arXiv:1207.4194] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    I. Amado, M. Kaminski and K. Landsteiner, Hydrodynamics of Holographic Superconductors, JHEP 05 (2009) 021 [arXiv:0903.2209] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    K. Maeda, M. Natsuume and T. Okamura, Universality class of holographic superconductors, Phys. Rev. D 79 (2009) 126004 [arXiv:0904.1914] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    X. Gao, A.M. Garcia-Garcia, H.B. Zeng and H.-Q. Zhang, Normal modes and time evolution of a holographic superconductor after a quantum quench, JHEP 06 (2014) 019 [arXiv:1212.1049] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    P. Basu, D. Das, S.R. Das and T. Nishioka, Quantum Quench Across a Zero Temperature Holographic Superfluid Transition, JHEP 03 (2013) 146 [arXiv:1211.7076] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    M. Rozali, D. Smyth, E. Sorkin and J.B. Stang, Holographic Stripes, Phys. Rev. Lett. 110 (2013) 201603 [arXiv:1211.5600] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. Donos and J.P. Gauntlett, Holographic striped phases, JHEP 08 (2011) 140 [arXiv:1106.2004] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    H. Liu, H. Ooguri, B. Stoica and N. Yunes, Spontaneous Generation of Angular Momentum in Holographic Theories, Phys. Rev. Lett. 110 (2013) 211601 [arXiv:1212.3666] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    A. Adams, P.M. Chesler and H. Liu, Holographic Vortex Liquids and Superfluid Turbulence, Science 341 (26 July 2013) 368 [arXiv:1212.0281] [INSPIRE].
  33. [33]
    V. Balasubramanian et al., Inhomogeneous holographic thermalization, JHEP 10 (2013) 082 [arXiv:1307.7086] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    P. Basu, D. Das, S.R. Das and K. Sengupta, Quantum Quench and Double Trace Couplings, JHEP 12 (2013) 070 [arXiv:1308.4061] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    L.N. Trefethen, Spectral methods in MATLAB, SIAM, Philadelphia, (2000).MATHCrossRefGoogle Scholar
  36. [36]
    T. Faulkner, G.T. Horowitz and M.M. Roberts, Holographic quantum criticality from multi-trace deformations, JHEP 04 (2011) 051 [arXiv:1008.1581] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Antonio M. García-García
    • 1
    • 2
  • Hua Bi Zeng
    • 2
    • 3
  • Hai-Qing Zhang
    • 2
  1. 1.Cavendish LaboratoryUniversity of CambridgeCambridgeU.K.
  2. 2.CFIF, Instituto Superior TécnicoUniversidade Técnica de LisboaLisboaPortugal
  3. 3.School of Mathematics and PhysicsBohai UniversityJinZhouChina

Personalised recommendations