Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory.

We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form ⟨W(t)VW(t)V⟩. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large c Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of ~t_{*}-(β/2π)logβ^{2}E_{w}E_{v}, where t_{*} is the fast scrambling time (β/2π)logc and E_{w},E_{v} are the energy scales of the W,V operators.

[1]  Thomas Hartman,et al.  Holographic entanglement entropy from 2d CFT: heavy states and local quenches , 2014, 1410.1392.

[2]  S. Shenker,et al.  Black holes and the butterfly effect , 2013, Journal of High Energy Physics.

[3]  H. Verlinde,et al.  Conformal bootstrap, universality and gravitational scattering , 2014, 1412.5205.

[4]  S. Shenker,et al.  Multiple shocks , 2013, 1312.3296.

[5]  M. Mattis.  Correlations in 2-dimensional critical theories , 1987 .

[6]  Deutsch,et al.  Quantum statistical mechanics in a closed system. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[7]  A. Polyakov,et al.  Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory - Nucl. Phys. B241, 333 (1984) , 1984 .

[8]  Omar Fawzi,et al.  Scrambling speed of random quantum circuits , 2012, 1210.6644.

[9]  G. Hooft,et al.  The gravitational shock wave of a massless particle , 1985 .

[10]  J. Maldacena,et al.  Time evolution of entanglement entropy from black hole interiors , 2013, 1303.1080.

[11]  P. Hayden,et al.  Black holes as mirrors: Quantum information in random subsystems , 2007, 0708.4025.

[12]  Stefan Leichenauer.  Disrupting Entanglement of Black Holes , 2014, 1405.7365.

[13]  Daniel A. Roberts,et al.  Localized shocks , 2014, 1409.8180.

[14]  J. Kaplan,et al.  Universality of long-distance AdS physics from the CFT bootstrap , 2014, 1403.6829.

[15]  Paul Ginsparg,et al.  Applied Conformal Field Theory , 1988, hep-th/9108028.

[16]  P. Hayden,et al.  Towards the fast scrambling conjecture , 2011, Journal of High Energy Physics.

[17]  L. Susskind.  Addendum to computational complexity and black hole horizons , 2014, 1403.5695.

[18]  O. Bohigas,et al.  Characterization of chaotic quantum spectra and universality of level fluctuation laws , 1984 .

[19]  J. Penedones,et al.  Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions , 2006, hep-th/0611123.

[20]  J. Penedones,et al.  Eikonal approximation in AdS/CFT: resumming the gravitational loop expansion , 2007, 0707.0120.

[21]  A. Larkin,et al.  Quasiclassical Method in the Theory of Superconductivity , 1969 .

[22]  L. Susskind,et al.  Switchbacks and the Bridge to Nowhere , 2014, 1408.2823.

[23]  Srednicki.  Chaos and quantum thermalization. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  Michael V Berry,et al.  Regular and irregular semiclassical wavefunctions , 1977 .

[25]  T. Takayanagi,et al.  Quantum entanglement of localized excited states at finite temperature , 2014, Journal of High Energy Physics.

[26]  A. Wightman,et al.  PCT, spin and statistics, and all that , 1964 .

[27]  Richard H. Price,et al.  Black Holes , 1997 .

[28]  Conformal four point functions and the operator product expansion , 2000, hep-th/0011040.

[29]  Eikonal approximation in AdS/CFT: from shock waves to four-point functions , 2006, hep-th/0611122.

[30]  L. Susskind,et al.  Complexity and Shock Wave Geometries , 2014, 1406.2678.

[31]  Thomas Hartman.  Entanglement Entropy at Large Central Charge , 2013, 1303.6955.

[32]  Black holes, shock waves, and causality in the AdS/CFT correspondence , 1999, hep-th/9901012.

[33]  Leonard Susskind,et al.  Entanglement is not enough , 2014, 1411.0690.

[34]  S. Shenker,et al.  Stringy effects in scrambling , 2014, 1412.6087.

[35]  A. Zamolodchikov.  Conformal symmetry in two dimensions: An explicit recurrence formula for the conformal partial wave amplitude , 1984 .

[36]  Eikonal approximation in AdS/CFT: resumming the gravitational loop expansion , 2007 .

[37]  L. Susskind.  Computational complexity and black hole horizons , 2014, 1402.5674.

[38]  M. Berry,et al.  Level clustering in the regular spectrum , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.