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Journal of High Energy Physics

, 2016:2 | Cite as

Butterflies from information metric

  • Masamichi MiyajiEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We study time evolution of distance between thermal states excited by local operators, with different external couplings. We find that growth of the distance implies growth of commutators of operators, signifying the local excitations are scrambled. We confirm this growth of distance by holographic computation, by evaluating volume of codimension 1 extremal volume surface. We find that the distance increases exponentially as \( {e}^{\frac{2\pi t}{\beta }} \). Our result implies that, in chaotic system, trajectories of excited thermal states exhibit high sensitivity to perturbation to the Hamiltonian, and the distance between them will be significant at the scrambling time. We also confirm the decay of two point function of holographic Wilson loops on thermofield double state.

Keywords

AdS-CFT Correspondence Black Holes Conformal Field Theory 1/N Expansion 

Notes

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.

References

  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, November 10, 2014.Google Scholar
  4. [4]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, JETP 28 (1969) 1200.Google Scholar
  8. [8]
    S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    P. Caputa, J. Simón, A. Štikonas, T. Takayanagi and K. Watanabe, Scrambling time from local perturbations of the eternal BTZ black hole, JHEP 08 (2015) 011 [arXiv:1503.08161] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
  14. [14]
    A. Kitaev, A simple model of quantum holography (part 2), Talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
  15. [15]
    D. Stanford, Many-body chaos at weak coupling, arXiv:1512.07687 [INSPIRE].
  16. [16]
    B. Michel, J. Polchinski, V. Rosenhaus and S.J. Suh, Four-point function in the IOP matrix model, JHEP 05 (2016) 048 [arXiv:1602.06422] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. Caputa, T. Numasawa and A. Veliz-Osorio, Scrambling without chaos in RCFT, arXiv:1602.06542 [INSPIRE].
  18. [18]
    Y. Gu and X.-L. Qi, Fractional Statistics and the Butterfly Effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].CrossRefGoogle Scholar
  19. [19]
    A.L. Fitzpatrick and J. Kaplan, A Quantum Correction To Chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate Operators and the 1/c Expansion: Lorentzian Resummations, High Order Computations and Super-Virasoro Blocks, arXiv:1606.02659 [INSPIRE].
  21. [21]
    S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. Polchinski, Chaos in the black hole S-matrix, arXiv:1505.08108 [INSPIRE].
  23. [23]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    G. Gur-Ari, M. Hanada and S.H. Shenker, Chaos in Classical D0-Brane Mechanics, JHEP 02 (2016) 091 [arXiv:1512.00019] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    E. Berkowitz, M. Hanada and J. Maltz, Chaos in Matrix Models and Black Hole Evaporation, arXiv:1602.01473 [INSPIRE].
  26. [26]
    B. Swingle, G. Bentsen, M. Schleier-Smith and P. Hayden, Measuring the scrambling of quantum information, arXiv:1602.06271 [INSPIRE].
  27. [27]
    E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, arXiv:1602.08272 [INSPIRE].
  28. [28]
    N. Sircar, J. Sonnenschein and W. Tangarife, Extending the scope of holographic mutual information and chaotic behavior, JHEP 05 (2016) 091 [arXiv:1602.07307] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic Publishers, Dordrecht, The Netherlands (1995).zbMATHGoogle Scholar
  31. [31]
    R. Jalabert and H. Pastawski, Environment-independent decoherence rate in classically chaotic systems, Phys. Rev. Lett. 86 (2001) 2490 [cond-mat/0010094].ADSCrossRefGoogle Scholar
  32. [32]
    P. Jacquod, P. Silvestrov and C. Beenakker, Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo, Phys. Rev. E 64 (2001) 055203(R) [nlin/0107044].ADSGoogle Scholar
  33. [33]
    N. Cerruti and S. Tomsovic, Sensitivity of Wave Field Evolution and Manifold Stability in Chaotic Systems, Phys. Rev. Lett. 88 (2002) 054103 [nlin/0108016].ADSCrossRefGoogle Scholar
  34. [34]
    T. Prosen and M. Znidaric, Stability of quantum motion and correlation decay, J. Phys. A 35 (2002) 1455 [nlin/0111014].ADSMathSciNetzbMATHGoogle Scholar
  35. [35]
    Z. Karkuszewski, C. Jarzynski and W. Zurek, Quantum Chaotic Environments, the Butterfly Effect, and Decoherence, Phys. Rev. Lett. 89 (2002) 170405 [quant-ph/0111002].ADSCrossRefGoogle Scholar
  36. [36]
    T. Gorin, T. Prosen and T.H. Seligman, A random matrix formulation of fidelity decay, New J. Phys. 6 (2004) 20 [nlin/0311022].ADSCrossRefGoogle Scholar
  37. [37]
    J. Emerson, Y. Weinstein, S. Lloyd and D. Cory, Fidelity Decay as an Efficient Indicator of Quantum Chaos, Phys. Rev. Lett. 89 (2002) 284102 [quant-ph/0207099].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    T. Gorin, T. Prosen, T. Seligman and M. Znidaric, Dynamics of Loschmidt echoes and fidelity decay, Phys. Rept. 435 (2006) 33 [quant-ph/0607050].ADSCrossRefGoogle Scholar
  39. [39]
    A. Goussev, R.A. Jalabert, H.M. Pastawski and D.A. Wisniacki, Loschmidt Echo, Scholarpedia 7 (2012) 11687 [arXiv:1206.6348].CrossRefGoogle Scholar
  40. [40]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  41. [41]
    H.P. Breuer, E.-M. Laine and J. Piilo, Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems, Phys. Rev. Lett. 103 (2009) 210401 [arXiv:0908.0238].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    P. Haikka, J. Goold, S. McEndoo, F. Plastina and S. Maniscalco, Non-Markovianity, Loschmidt echo, and criticality: A unified picture, Phys. Rev. A 85 (2012) 060101 [arXiv:1202.2997].ADSCrossRefGoogle Scholar
  43. [43]
    H.T. Quan, Z. Song, X.F. Liu, P. Zanardi and C.P. Sun, Decay of Loschmidt Echo Enhanced by Quantum Criticality, Phys. Rev. Lett. 96 (2006) 140604 [quant-ph/0509007].ADSCrossRefGoogle Scholar
  44. [44]
    P. Zanardi and N. Paunkovic, Ground state overlap and quantum phase transitions, Phys. Rev. E 74 (2006) 031123 [quant-ph/0512249].ADSMathSciNetGoogle Scholar
  45. [45]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    D. Bak, M. Gutperle and S. Hirano, A dilatonic deformation of AdS 5 and its field theory dual, JHEP 05 (2003) 072 [hep-th/0304129] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    D. Bak, M. Gutperle and S. Hirano, Three dimensional Janus and time-dependent black holes, JHEP 02 (2007) 068 [hep-th/0701108] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  50. [50]
    M. Miyaji, S. Ryu, T. Takayanagi and X. Wen, Boundary States as Holographic Duals of Trivial Spacetimes, JHEP 05 (2015) 152 [arXiv:1412.6226] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
  52. [52]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan

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