Skip to main content
SpringerLink
Log in

Decay of a thermofield-double state in chaotic quantum systems

From random matrices to spin systems

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Hayden, J. Preskill, J. High Energy Phys. 0709, 120 (2007)

    Article  ADS  Google Scholar 

  2. Y. Sekino, L. Susskind, J. High Energy Phys. 0810, 065 (2008)

    Article  ADS  Google Scholar 

  3. N. Lashkari, D. Stanford, M. Hastings, T. Osborne, P. Hayden, J. High Energy Phys. 1304, 022 (2013)

    Article  ADS  Google Scholar 

  4. J.L.F. Barbon, E. Rabinovici, J. High Energy Phys. 0311, 047 (2003)

    Article  ADS  Google Scholar 

  5. J.L.F. Barbon, E. Rabinovici, Fortsch. Phys. 52, 642 (2004)

    Article  ADS  Google Scholar 

  6. K. Papadodimas, S. Raju, Phys. Rev. Lett. 115, 211601 (2015)

    Article  ADS  Google Scholar 

  7. O. Aharony, S.S. Gubser, J.M Maldacena, H. Ooguri, Y. Oz, Phys. Rep. 323, 183 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  8. J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  9. S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. B428, 105 (1998)

    Article  ADS  Google Scholar 

  10. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  11. J. Maldacena, S.H. Shenker, D. Stanford, J. High Energy Phys. 08, 106 (2016)

    Article  ADS  Google Scholar 

  12. S.H. Shenker, D. Stanford, J. High Energy Phys. 05, 132 (2015)

    Article  ADS  Google Scholar 

  13. N. Tsuji, T. Shitara, M. Ueda, https://doi.org/arXiv:1706.09160 (2017)

  14. J.M. Maldacena, J. High Energy Phys. 04, 021 (2003)

    Article  ADS  Google Scholar 

  15. A. del Campo, J. Molina-Vilaplana, J. Sonner, Phys. Rev. D 95, 126008 (2017)

    Article  ADS  Google Scholar 

  16. E. Dyer, G. Gur-Ari, J. High Energy Phys 08, 075 (2017)

    Article  ADS  Google Scholar 

  17. J.S. Cotler, Guy Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker, D. Stanford, A. Streicher, M. Tezuka, J. High Energy Phys. 05, 118 (2017)

    Article  ADS  Google Scholar 

  18. J. Cotler, N. Hunter-Jones, J. Liu, B. Yoshida, J. High Energy Phys. 11, 48 (2017)

    Article  ADS  Google Scholar 

  19. L. Leviandier, M. Lombardi, R. Jost, J.P. Pique, Phys. Rev. Lett. 56, 2449 (1986)

    Article  ADS  Google Scholar 

  20. T. Guhr, H. Weidenmüller, J. Chem. Phys. 146, 21 1990)

    Google Scholar 

  21. Y. Alhassid, R. D.Levine, Phys. Rev. A 46, 4650 (1992)

    Article  ADS  Google Scholar 

  22. T. Gorin, T.H. Seligman, Phys. Rev. E 65, 026214 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  23. E.J. Torres-Herrera, L.F. Santos, Philos. Trans. R. Soc. A 375, 20160434 (2017)

    Article  ADS  Google Scholar 

  24. E.J. Torres-Herrera, A. García-García, L.F. Santos, Phys. Rev. B97, 060303(R) (2018)

    Article  ADS  Google Scholar 

  25. N.Y. Yao, F. Grusdt, B. Swingle, M.D. Lukin, D.M. Stamper-Kurn, J.E. Moore, E.A. Demler, https://doi.org/arXiv:1607.01801 (2016)

  26. B. Swingle, G. Bentsen, M. Schleier-Smith, P. Hayden, Phys. Rev. A 94, 040302 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  27. L. García-Álvarez, I.L. Egusquiza, L. Lamata, A. del Campo, J. Sonner, E. Solano, Phys. Rev. Lett. 119, 040501 (2017)

    Article  ADS  Google Scholar 

  28. M. Gärttner, J.G. Bohnet, A. Safavi-Naini, M.L. Wall, J.J. Bollinger, A.M. Rey, Nat. Phys. 13, 781 (2017)

    Article  Google Scholar 

  29. J. Li, R.Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, J. Du, Phys. Rev. X 7, 031011 (2017)

    Google Scholar 

  30. M.L. Mehta, Random Matrices, 3rd edn. (Elsevier, San Diego, 2004)

  31. T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981)

    Article  ADS  Google Scholar 

  32. A. Kitaev, Talks at KITP, April 7, 2015 https://doi.org/online.kitp.ucsb.edu/online/entangled15/ kitaev/, and May 27, 2015 https://doi.org/online.kitp.ucsb.edu/online/entangled15/kitaev2/ April 7, 2015, https://doi.org/online.kitp.ucsb.edu/online/entangled15/kitaev/, and May 27, 2015, https://doi.org/online.kitp.ucsb.edu/online/entangled15/kitaev2/

  33. S. Sachdev, Phys. Rev. X 5, 041025 (2015)

    Google Scholar 

  34. J. Sonner, M. Vielma, J. High. Energy Phys. 11, 149 (2017)

    Article  ADS  Google Scholar 

  35. I. Danshita, M. Hanada, M. Tezuka, Prog. Theor. Exp. Phys. 8, 083I01 (2017)

    Google Scholar 

  36. M. Schreiber, S.S. Hodgman, P. Bordia, H.P. Lüschen, M.H. Fischer, R. Vosk, E. Altman, U. Schneider, I. Bloch, Science 349, 842 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  37. C.B. Chiu, E.C.G. Sudarshan, B. Misra Phys. Rev. D 16, 520 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  38. L. Fonda, G.C. Ghirardi, A. Rimini, Rep. Prog. Phys. 41, 587 (1978)

    Article  ADS  Google Scholar 

  39. L.A. Khalfin, Sov. Phys. JETP 6, 1053 (1958) [Zh. Eks. Teor. Fiz. 33, 1371 (1957)]

    ADS  Google Scholar 

  40. P.L. Knight, Phys. Lett. A 61, 25 (1977)

    Article  ADS  Google Scholar 

  41. J. Martorell, J.G. Muga, D.W.L. Sprung, Lect. Notes Phys. 789, 239 (2009)

    Article  ADS  Google Scholar 

  42. K. Urbanowski, Eur. Phys. J. D 54, 25 (2009)

    Article  ADS  Google Scholar 

  43. A. del Campo, Phys. Rev. A 84, 012113 (2011)

    Article  ADS  Google Scholar 

  44. E.J. Torres-Herrera, L.F. Santos, Phys. Rev. A 89, 043620 (2014)

    Article  ADS  Google Scholar 

  45. E.J. Torres-Herrera, M. Vyas, L.F. Santos, New J. Phys. 16, 063010 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  46. E.J. Torres-Herrera, L.F. Santos, Phys. Rev. A 90, 033623 (2014)

    Article  ADS  Google Scholar 

  47. E.J. Torres-Herrera, L.F. Santos, Phys. Rev. E 89, 062110 (2014)

    Article  ADS  Google Scholar 

  48. A. del Campo, New J. Phys. 18, 015014 (2016)

    Article  ADS  Google Scholar 

  49. E.R. Fiori, H.M. Pastawski, Chem. Phys. Lett. 420, 35 (2006)

    Article  ADS  Google Scholar 

  50. I. Ersak, Sov. J. Nucl. Phys. 9, 263 (1969)

    Google Scholar 

  51. M. Beau, J. Kiukas, I.L. Egusquiza, A. del Campo, Phys. Rev. Lett. 119, 130401 (2017)

    Article  ADS  Google Scholar 

  52. V.A. Fock, S.N. Krylov, Zh. Eksp. Teor. Fiz. 17, 93 (1947)

    Google Scholar 

  53. E.J. Torres-Herrera, D. Kollmar, L.F. Santos, Phys. Scr. T 165, 014018 (2015)

    Article  ADS  Google Scholar 

  54. M. Távora, E.J. Torres-Herrera, L.F. Santos, Phys. Rev. A 94, 041603 (2016)

    Article  ADS  Google Scholar 

  55. M. Távora, E.J. Torres-Herrera, L.F. Santos, Phys. Rev. A 95, 013604 (2017)

    Article  ADS  Google Scholar 

  56. A. Chenu, J. Molina-Vilaplana, A. del Campo, https://doi.org/arXiv:1804.09188 (2018)

  57. E. Brézin, S. Hikami, Phys. Rev. E 55, 4067 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  58. A.V. Sologubenko, K. Giannóo, H.R. Ott, U. Ammerahl, A. Revcolevschi, Phys. Rev. Lett. 84, 2714 (2000)

    Article  ADS  Google Scholar 

  59. N. Hlubek, P. Ribeiro, R. Saint-Martin, A. Revcolevschi, G. Roth, G. Behr, B. Büchner, C. Hess, Phys. Rev. B 81, 020405(R) (2010)

    Article  ADS  Google Scholar 

  60. K.X. Wei, C. Ramanathan, P. Cappellaro, Phys. Rev. Lett. 120, 070501 (2018)

    Article  ADS  Google Scholar 

  61. P. Jurcevic, B.P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, C.F. Roos, Nature 511, 202 (2014)

    Article  ADS  Google Scholar 

  62. P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A.V. Gorshkov, C. Monroe, Nature 511, 198 (2014)

    Article  ADS  Google Scholar 

  63. S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnorrberger, A.M. Rey, A. Polkovnikov, E.A. Demler, M.D. Lukin, I. Bloch, Science 319, 295 (2008)

    Article  ADS  Google Scholar 

  64. L.F. Santos, G. Rigolin, C.O. Escobar, Phys. Rev. A 69, 042304 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  65. R. Nandkishore, D. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015)

    Article  ADS  Google Scholar 

  66. D.J. Luitz, Y.B. Lev, Ann. Phys. (Berlin) 529, 1600350 (2017)

    Article  ADS  Google Scholar 

  67. H.A. Bethe, Z. Phys. 71, 205 (1931)

    Article  ADS  Google Scholar 

  68. L.F. Santos, J. Phys. A 37, 4723 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  69. F. Dukesz, M. Zilbergerts, L.F. Santos, New J. Phys. 11, 043026 (2009)

    Article  ADS  Google Scholar 

  70. E.J. Torres-Herrera, L.F. Santos, Ann. Phys. (Berlin) 529, 1600284 (2017)

    Article  ADS  Google Scholar 

  71. E.J. Torres-Herrera, L.F. Santos, Phys. Rev. B 92, 014208 (2015)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Massachusetts, Boston, MA, 02125, USA

    A. del Campo

  2. Department of Systems Engineering, Technical University of Cartagena, C/Dr Fleming S/N, 30202, Cartagena, Spain

    J. Molina-Vilaplana

  3. Department of Physics, Yeshiva University, New York, NA, 10016, USA

    L. F. Santos

  4. Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211, Genève 4, Switzerland

    J. Sonner

Authors
  1. A. del Campo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Molina-Vilaplana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. F. Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Sonner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Molina-Vilaplana.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

del Campo, A., Molina-Vilaplana, J., Santos, L.F. et al. Decay of a thermofield-double state in chaotic quantum systems. Eur. Phys. J. Spec. Top. 227, 247–258 (2018). https://doi.org/10.1140/epjst/e2018-00083-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2018-00083-5

Search

Navigation