Advertisement

Journal of High Energy Physics

, 2019:18 | Cite as

Spectral representation of thermal OTO correlators

  • Soumyadeep ChaudhuriEmail author
  • Chandramouli Chowdhury
  • R. Loganayagam
Open Access
Regular Article - Theoretical Physics
  • 106 Downloads

Abstract

We study the spectral representation of finite temperature, out of time ordered (OTO) correlators on the multi-time-fold generalised Schwinger-Keldysh contour. We write the contour-ordered correlators as a sum over time-order permutations acting on a fundamental array of Wightman correlators. We decompose this Wightman array in a basis of column vectors, which provide a natural generalisation of the familiar retarded-advanced basis in the finite temperature Schwinger-Keldysh formalism. The coefficients of this decomposition take the form of generalised spectral functions, which are Fourier transforms of nested and double commutators. Our construction extends a variety of classical results on spectral functions in the SK formalism at finite temperature to the OTO case.

Keywords

Thermal Field Theory Quantum Dissipative Systems Nonperturbative Effects 

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.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [INSPIRE].Google Scholar
  3. [3]
    R.P. Feynman and F.L. Vernon Jr., The Theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24 (1963) 118 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu, Equilibrium and Nonequilibrium Formalisms Made Unified, Phys. Rept. 118 (1985) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E.A. Calzetta and B.-L.B. Hu, Nonequilibrium Quantum Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2008).Google Scholar
  6. [6]
    A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press (2011).Google Scholar
  7. [7]
    A. Kamenev and A. Levchenko, Keldysh technique and nonlinear σ-model: Basic principles and applications, Adv. Phys. 58 (2009) 197 [arXiv:0901.3586] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    G. Stefanucci and R. Van Leeuwen, Nonequilibrium many-body theory of quantum systems: a modern introduction, Cambridge University Press (2013).Google Scholar
  9. [9]
    L.M. Sieberer, M. Buchhold and S. Diehl, Keldysh Field Theory for Driven Open Quantum Systems, Rept. Prog. Phys. 79 (2016) 096001 [arXiv:1512.00637] [INSPIRE].
  10. [10]
    F.M. Haehl, R. Loganayagam and M. Rangamani, Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace, JHEP 06 (2017) 069 [arXiv:1610.01940] [INSPIRE].
  11. [11]
    A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, Sov. J. Exp. Theor. Phys. 28 (1969) 1200.ADSGoogle Scholar
  12. [12]
    I.L. Aleiner, L. Faoro and L.B. Ioffe, Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves, Annals Phys. 375 (2016) 378 [arXiv:1609.01251] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    F.M. Haehl, R. Loganayagam, P. Narayan and M. Rangamani, Classification of out-of-time-order correlators, SciPost Phys. 6 (2019) 001 [arXiv:1701.02820] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. Fan, P. Zhang, H. Shen and H. Zhai, Out-of-Time-Order Correlation for Many-Body Localization, arXiv:1608.01914 [INSPIRE].
  17. [17]
    Y. Chen, Quantum Logarithmic Butterfly in Many Body Localization, arXiv:1608.02765 [INSPIRE].
  18. [18]
    B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95 (2017) 060201 [arXiv:1608.03280] [INSPIRE].
  19. [19]
    Y. Huang, Y. Zhang and X. Chen, Out-of-time-ordered correlators in many-body localized systems, Annalen Phys. 529 (2017) 1600318 [arXiv:1608.01091] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    N. Tsuji, T. Shitara and M. Ueda, Out-of-time-order fluctuation-dissipation theorem, Phys. Rev. E 97 (2018) 012101 [arXiv:1612.08781] [INSPIRE].
  23. [23]
    N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95 (2017) 012120 [arXiv:1609.00015] [INSPIRE].
  24. [24]
    N. Yunger Halpern, B. Swingle and J. Dressel, Quasiprobability behind the out-of-time-ordered correlator, Phys. Rev. A 97 (2018) 042105 [arXiv:1704.01971] [INSPIRE].
  25. [25]
    S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    L. Iliesiu, M. Koloğlu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin, The Conformal Bootstrap at Finite Temperature, JHEP 10 (2018) 070 [arXiv:1802.10266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10 (2018) 127 [arXiv:1801.00010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    F.M. Haehl and M. Rozali, Fine Grained Chaos in AdS 2 Gravity, Phys. Rev. Lett. 120 (2018) 121601 [arXiv:1712.04963] [INSPIRE].
  30. [30]
    F.M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    P. Basu and K. Jaswin, Higher point OTOCs and the bound on chaos, arXiv:1809.05331 [INSPIRE].
  32. [32]
    F.M. Haehl, R. Loganayagam, P. Narayan, A.A. Nizami and M. Rangamani, Thermal out-of-time-order correlators, KMS relations and spectral functions, JHEP 12 (2017) 154 [arXiv:1706.08956] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    L. Foini and J. Kurchan, The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators, arXiv:1803.10658 [INSPIRE].
  34. [34]
    G. Zhu, M. Hafezi and T. Grover, Measurement of many-body chaos using a quantum clock, Phys. Rev. A 94 (2016) 062329 [arXiv:1607.00079] [INSPIRE].
  35. [35]
    M. Gärttner, J.G. Bohnet, A. Safavi-Naini, M.L. Wall, J.J. Bollinger and A.M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet, Nature Phys. 13 (2017) 781 [arXiv:1608.08938] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    S. Weinberg, The Quantum theory of fields. Volume 1: Foundations, Cambridge University Press (2005).Google Scholar
  37. [37]
    T.S. Evans, Three Point Functions At Finite Temperature, Phys. Lett. B 249 (1990) 286 [INSPIRE].
  38. [38]
    T.S. Evans, Spectral representation of three point functions at finite temperature, Phys. Lett. B 252 (1990) 108 [INSPIRE].
  39. [39]
    T.S. Evans, N-point finite temperature expectation values at real times, Nucl. Phys. B 374 (1992) 340 [INSPIRE].
  40. [40]
    J.C. Taylor, Spectral representation of hard thermal loops, Phys. Rev. D 48 (1993) 958 [INSPIRE].
  41. [41]
    M.E. Carrington and U.W. Heinz, Three point functions at finite temperature, Eur. Phys. J. C 1 (1998) 619 [hep-th/9606055] [INSPIRE].
  42. [42]
    D.-f. Hou, E. Wang and U.W. Heinz, n-point functions at finite temperature, J. Phys. G 24 (1998) 1861 [hep-th/9807118] [INSPIRE].
  43. [43]
    E. Wang and U.W. Heinz, A Generalized fluctuation dissipation theorem for nonlinear response functions, Phys. Rev. D 66 (2002) 025008 [hep-th/9809016] [INSPIRE].
  44. [44]
    D.-f. Hou, M.E. Carrington, R. Kobes and U.W. Heinz, Four-point spectral functions and Ward identities in hot QED, Phys. Rev. D 61 (2000) 085013 [Erratum ibid. D 67 (2003) 049902] [hep-ph/9911494] [INSPIRE].
  45. [45]
    F. Guerin, Retarded-advanced N-point Green functions in thermal field theories, Nucl. Phys. B 432 (1994) 281 [hep-ph/9306210] [INSPIRE].
  46. [46]
    F. Guerin, Four point functions in Keldysh basis, hep-ph/0105313 [INSPIRE].
  47. [47]
    H. Chu and H. Umezawa, Time ordering theorem and calculational recipes for thermal field dynamics, Phys. Lett. A 177 (1993) 385 [INSPIRE].
  48. [48]
    P.A. Henning, The Column vector calculus for thermo field dynamics of relativistic quantum fields, Phys. Lett. B 313 (1993) 341 [nucl-th/9305007] [INSPIRE].
  49. [49]
    P.A. Henning, Thermo field dynamics for quantum fields with continuous mass spectrum, Phys. Rept. 253 (1995) 235 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M.E. Carrington, T. Fugleberg, D.S. Irvine and D. Pickering, Real time statistical field theory, Eur. Phys. J. C 50 (2007) 711 [hep-ph/0608298] [INSPIRE].
  51. [51]
    E. Braaten and R.D. Pisarski, Simple effective Lagrangian for hard thermal loops, Phys. Rev. D 45 (1992) R1827 [INSPIRE].
  52. [52]
    R. Baier and A. Niegawa, Analytic continuation of thermal N point functions from imaginary to real energies, Phys. Rev. D 49 (1994) 4107 [hep-ph/9307362] [INSPIRE].
  53. [53]
    P. Aurenche and T. Becherrawy, A Comparison of the real time and the imaginary time formalisms of finite temperature field theory for 2, 3 and 4 point Green’s functions, Nucl. Phys. B 379 (1992) 259 [INSPIRE].
  54. [54]
    M.A. van Eijck and C.G. van Weert, Finite temperature retarded and advanced Green functions, Phys. Lett. B 278 (1992) 305 [INSPIRE].
  55. [55]
    M.A. van Eijck, R. Kobes and C.G. van Weert, Transformations of real time finite temperature Feynman rules, Phys. Rev. D 50 (1994) 4097 [hep-ph/9406214] [INSPIRE].
  56. [56]
    R.L. Kobes and G.W. Semenoff, Discontinuities of Green Functions in Field Theory at Finite Temperature and Density, Nucl. Phys. B 260 (1985) 714 [INSPIRE].
  57. [57]
    R.L. Kobes and G.W. Semenoff, Discontinuities of Green Functions in Field Theory at Finite Temperature and Density. 2, Nucl. Phys. B 272 (1986) 329 [INSPIRE].
  58. [58]
    R. Kobes, Retarded functions, dispersion relations and Cutkosky rules at zero and finite temperature, Phys. Rev. D 43 (1991) 1269 [INSPIRE].
  59. [59]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
  60. [60]
    R. Omnés, On Locality, Growth and Transport of Entanglement, arXiv:1212.0331.
  61. [61]
    H. Kim and D.A. Huse, Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System, Phys. Rev. Lett. 111 (2013) 127205 [arXiv:1306.4306].ADSCrossRefGoogle Scholar
  62. [62]
    D. Stanford, Many-body chaos at weak coupling, JHEP 10 (2016) 009 [arXiv:1512.07687] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    S. Caron-Huot, Hard thermal loops in the real-time formalism, JHEP 04 (2009) 004 [arXiv:0710.5726] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    C.P. Herzog and D.T. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality, Phys. Rev. Lett. 101 (2008) 081601 [arXiv:0805.0150] [INSPIRE].
  66. [66]
    S. Chaudhuri and R. Loganayagam, Simplifying OTO Diagrammatics, to appear.Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Soumyadeep Chaudhuri
    • 1
    Email author
  • Chandramouli Chowdhury
    • 1
    • 2
  • R. Loganayagam
    • 1
  1. 1.International Centre for Theoretical Sciences (ICTS-TIFR)Tata Institute of Fundamental ResearchBangaloreIndia
  2. 2.Department of PhysicsIndian Institute of Technology GuwahatiGuwahatiIndia

Personalised recommendations