Holographic relaxation of finite size isolated quantum systems

  • Javier Abajo-Arrastia
  • Emilia da Silva
  • Esperanza Lopez
  • Javier MasEmail author
  • Alexandre Serantes
Open Access


We study holographically the out of equilibrium dynamics of a finite size closed quantum system in 2 + 1 dimensions, modelled by the collapse of a shell of a massless scalar field in AdS4. In global coordinates there exists a variety of evolutions towards final black hole formation which we relate with different patterns of relaxation in the dual field theory. For large scalar initial data rapid thermalization is achieved as a priori expected. Interesting phenomena appear for small enough amplitudes. Such shells do not generate a black hole by direct collapse, but quite generically, an apparent horizon emerges after enough bounces off the AdS boundary. We relate this bulk evolution with relaxation processes at strong coupling which delay in reaching an ergodic stage. Besides the dynamics of bulk fields, we monitor the entanglement entropy, finding that it oscillates quasi-periodically before final equilibration. The radial position of the travelling shell is brought in correspondence with the evolution of the pattern of entanglement in the dual field theory. We propose, thereafter, that the observed oscillations are the dual counterpart of the quantum revivals studied in the literature. The entanglement entropy is not only able to portrait the streaming of entangled excitations, but it is also a useful probe of interaction effects.


Gauge-gravity correspondence Black Holes in String Theory 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]
    A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 (2011) 863 [arXiv:1007.5331] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, Document LA-1940 (1955).Google Scholar
  3. [3]
    G.P. Berman and F.M. Izrailev, The Fermi-Pasta-Ulam problem: Fifty years of progress, Chaos 15 (2005) 015104 [nlin/0411062].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T. Kinoshita, T. Wenger and D.S. Weiss, A quantum Newtons cradle, Nature 440 (2006) 900.ADSCrossRefGoogle Scholar
  5. [5]
    M. Gring, M. Kuhnert, T. Langen, T. Kitazgawa, B. Rauer, M. Schreitl et al., Relaxation and Prethermalization in an Isolated Quantum System, Science 337 (2012) 1318 [arXiv:1112.0013].ADSCrossRefGoogle Scholar
  6. [6]
    S. Trotzky, Y.-A. Chen, A. Flesch, I. P Mc Cullough, U. Schollwöck, J. Eisert et al., Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas, Nature Phys. 8 (2012) 325 [arXiv:1101.2659].ADSCrossRefGoogle Scholar
  7. [7]
    E.T. Jaynes, Information Theory and Statistical Mechanics, Phys. Rev. 106 (1957) 620 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States in 1D Lattice Hard-Core Bosons, Phys. Rev. Lett. 98 (2007) 050405 [cond-mat/0604476].ADSCrossRefGoogle Scholar
  9. [9]
    M. Kollar, F.A. Wolf and M. Eckstein, Generalized Gibbs ensemble predicction of prethermalization plateaus and their relation to non-thermal steady states in integrable systems, Phys. Rev. B 84 (2011) 054304 [arXiv:1102.2117].ADSCrossRefGoogle Scholar
  10. [10]
    D.A. Smith, M. Gring, T. Langen, M. Kuhnert, B. Rauer et al., Prethermalization Revealed by the Relaxation Dynamics of Full Distribution Functions, New J. Phys. 15 (2013) 075011 [arXiv:1212.4645] [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    G. Mussardo, Infinite-time Average of Local Fields in an Integrable Quantum Field Theory after a Quantum Quench, Phys. Rev. Lett. 111 (2013) 100401 [arXiv:1308.4551] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: a conformal field theory approach, J. Stat. Mech.: Theo. Exp. 2007 (2007) P10004 [arXiv:0708.3750].CrossRefGoogle Scholar
  17. [17]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 2005 (2005) P04010 [cond-mat/0503393] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Albash and C.V. Johnson, Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches, New J. Phys. 13 (2011) 045017 [arXiv:1008.3027] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].ADSGoogle Scholar
  22. [22]
    M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    T. Takayanagi and T. Ugajin, Measuring Black Hole Formations by Entanglement Entropy via Coarse-Graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    F. Pretorius and M.W. Choptuik, Gravitational collapse in (2 + 1)-dimensional AdS space-time, Phys. Rev. D 62 (2000) 124012 [gr-qc/0007008] [INSPIRE].ADSGoogle Scholar
  25. [25]
    M.W. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett. 70 (1993) 9 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    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
  27. [27]
    M. Greiner, O.Mandel, T.W. Hänsh and I. Bolch, Collapse and revival of the matter wave field of a Bose-Einstein condensate, Nature 419 (2002) 51.ADSCrossRefGoogle Scholar
  28. [28]
    H. Rieger and F. Iglói, Quantum relaxation after a quench in systems with boundaries, Phys. Rev. Lett. 106 (2011) 035701 [arXiv:1011.3664].ADSCrossRefGoogle Scholar
  29. [29]
    J. Häppölä, G.B. Halász and A. Hamma, Revivals of a closed quantum system and Lieb-Robinson speed, Phys. Rev. A 85 (2012) 032114.ADSCrossRefGoogle Scholar
  30. [30]
    J. Cardy, Thermalization and Revivals after a Quantum Quench in Conformal Field Theory, arXiv:1403.3040 [INSPIRE].
  31. [31]
    J. Jalmuzna, A. Rostworowski and P. Bizon, A Comment on AdS collapse of a scalar field in higher dimensions, Phys. Rev. D 84 (2011) 085021 [arXiv:1108.4539] [INSPIRE].ADSGoogle Scholar
  32. [32]
    O.J.C. Dias, G.T. Horowitz and J.E. Santos, Gravitational Turbulent Instability of Anti-de Sitter Space, Class. Quant. Grav. 29 (2012) 194002 [arXiv:1109.1825] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D. Garfinkle and L.A. Pando Zayas, Rapid Thermalization in Field Theory from Gravitational Collapse, Phys. Rev. D 84 (2011) 066006 [arXiv:1106.2339] [INSPIRE].ADSGoogle Scholar
  34. [34]
    D. Garfinkle, L.A. Pando Zayas and D. Reichmann, On Field Theory Thermalization from Gravitational Collapse, JHEP 02 (2012) 119 [arXiv:1110.5823] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    A. Buchel, L. Lehner and S.L. Liebling, Scalar Collapse in AdS, Phys. Rev. D 86 (2012) 123011 [arXiv:1210.0890] [INSPIRE].ADSGoogle Scholar
  36. [36]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    M. Maliborski and A. Rostworowski, Time-Periodic Solutions in an Einstein AdS-Massless-Scalar-Field System, Phys. Rev. Lett. 111 (2013) 051102 [arXiv:1303.3186] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A. Buchel, S.L. Liebling and L. Lehner, Boson stars in AdS spacetime, Phys. Rev. D 87 (2013) 123006 [arXiv:1304.4166] [INSPIRE].ADSGoogle Scholar
  39. [39]
    M. Maliborski and A. Rostworowski, A comment onBoson stars in AdS”, arXiv:1307.2875 [INSPIRE].
  40. [40]
    P. Bizon and J. Jalmużna, Globally regular instability of AdS 3, Phys. Rev. Lett. 111 (2013) 041102 [arXiv:1306.0317] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M. Maliborski, Instability of Flat Space Enclosed in a Cavity, Phys. Rev. Lett. 109 (2012) 221101 [arXiv:1208.2934] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M.P. Heller, R.A. Janik and P. Witaszczyk, The characteristics of thermalization of boost-invariant plasma from holography, Phys. Rev. Lett. 108 (2012) 201602 [arXiv:1103.3452] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M.P. Heller, D. Mateos, W. van der Schee and D. Trancanelli, Strong Coupling Isotropization of Non-Abelian Plasmas Simplified, Phys. Rev. Lett. 108 (2012) 191601 [arXiv:1202.0981] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    O.J.C. Dias, G.T. Horowitz and J.E. Santos, Black holes with only one Killing field, JHEP 07 (2011) 115 [arXiv:1105.4167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    V.I. Yudovihc, On the loss of smoothness of the solutions of Eulers equations with time in Russian, Dinamika Sploshn. Sredy 16 (1974) 71.Google Scholar
  46. [46]
    G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].ADSMathSciNetGoogle Scholar
  47. [47]
    E. Berti, V. Cardoso and P. Pani, Breit-Wigner resonances and the quasinormal modes of anti-de Sitter black holes, Phys. Rev. D 79 (2009) 101501 [arXiv:0903.5311] [INSPIRE].ADSMathSciNetGoogle Scholar
  48. [48]
    M.A. Cazalilla, Effect of Suddenly Turning on Interactions in the Luttinger Model, Phys. Rev. Lett. 97 (2006) 156403.ADSCrossRefGoogle Scholar
  49. [49]
    W.H. Press, S.A. Teutolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes, Cambridge University Press, section 17.3.Google Scholar
  50. [50]
    C.V. Johnson, Large-N Phase Transitions, Finite Volume and Entanglement Entropy, JHEP 03 (2014) 047 [arXiv:1306.4955] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    W.R. Kelly and A.C. Wall, Coarse-grained entropy and causal holographic information in AdS/CFT, JHEP 03 (2014) 118 [arXiv:1309.3610] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
  56. [56]
    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
  57. [57]
    N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglementthermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  59. [59]
    M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of Entanglement Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    G. Vidal, Entanglement Renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement Renormalization for Quantum Fields in Real Space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    V.E. Hubeny and M. Rangamani, Unstable horizons, JHEP 05 (2002) 027 [hep-th/0202189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Javier Abajo-Arrastia
    • 1
  • Emilia da Silva
    • 1
  • Esperanza Lopez
    • 1
  • Javier Mas
    • 2
    Email author
  • Alexandre Serantes
    • 2
  1. 1.Instituto de Física Teórica IFT UAM/CSIC, C-XVIUniversidad Autónoma de MadridMadridSpain
  2. 2.Departamento de Física de PartículasUniversidade de Santiago de Compostela, and Instituto Galego de Física de Altas Enerxías IGFAESantiago de CompostelaSpain

Personalised recommendations