A bound on chaos

Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas

doi:10.1007/JHEP08(2016)106

Title: A bound on chaos

Abstract

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

Authors:

Maldacena, Juan ^[1]; Shenker, Stephen H. ^[2]; Stanford, Douglas ^[1]

Princeton Univ., NJ (United States)
Stanford Univ., CA (United States)

Publication Date:: Wed Aug 17 00:00:00 EDT 2016

Research Org.:: Princeton Univ., NJ (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1434623

Grant/Contract Number:: SC0009988

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 8; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 1/N Expansion; Black Holes; AdS-CFT Correspondence

Citation Formats


                    Maldacena, Juan, Shenker, Stephen H., and Stanford, Douglas. A bound on chaos.  United States: N. p., 2016. 
Web.  doi:10.1007/JHEP08(2016)106.

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                    Maldacena, Juan, Shenker, Stephen H., & Stanford, Douglas. A bound on chaos.  United States.  https://doi.org/10.1007/JHEP08(2016)106

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                    Maldacena, Juan, Shenker, Stephen H., and Stanford, Douglas. Wed .  
"A bound on chaos".  United States.  https://doi.org/10.1007/JHEP08(2016)106.  https://www.osti.gov/servlets/purl/1434623.

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@article{osti_1434623,

  title        = {A bound on chaos},

  author       = {Maldacena, Juan and Shenker, Stephen H. and Stanford, Douglas},

  abstractNote = {We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.},

  doi          = {10.1007/JHEP08(2016)106},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 8,

  volume       = 2016,

  place        = {United States},

  year         = {Wed Aug 17 00:00:00 EDT 2016},

  month        = {Wed Aug 17 00:00:00 EDT 2016}

}

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Seshadri, Akshay; Madhok, Vaibhav; Lakshminarayan, Arul
Physical Review E, Vol. 98, Issue 5
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Chaotic transport of electron wave packet in Weyl semimetal slab
journal, May 2019

Yar, Abdullah; Ullah Khan, Safi
Physical Review E, Vol. 99, Issue 5
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Finite-Size Scaling of Out-of-Time-Ordered Correlators at Late Times
journal, July 2019

Huang, Yichen; Brandão, Fernando G. S. L.; Zhang, Yong-Liang
Physical Review Letters, Vol. 123, Issue 1
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Quantum localization bounds Trotter errors in digital quantum simulation
journal, April 2019

Heyl, Markus; Hauke, Philipp; Zoller, Peter
Science Advances, Vol. 5, Issue 4
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A traffic data collection and analysis method based on wireless sensor network
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Wang, Huan; Ouyang, Min; Meng, Qingyuan
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Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems
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Chenu, Aurélia; Molina-Vilaplana, Javier; del Campo, Adolfo
Quantum, Vol. 3
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Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators
text, January 2018

Rammensee, Josef; Urbina, Juan Diego; Richter, Klaus
Universität Regensburg
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Reversible Quantum Information Spreading in Many-Body Systems near Criticality
text, January 2019

Hummel, Quirin; Geiger, Benjamin; Urbina, Juan Diego
Universität Regensburg
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Probing typical black hole microstates
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de Boer, Jan; van Breukelen, Rik; Lokhande, Sagar F.
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Nearly AdS2 holography in quantum CGHS model
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Hirano, Shinji; Lei, Yang
Journal of High Energy Physics, Vol. 2020, Issue 1
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Entanglement and chaos in warped conformal field theories
journal, April 2019

Apolo, Luis; He, Song; Song, Wei
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Black holes and random matrices
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Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori
Journal of High Energy Physics, Vol. 2017, Issue 5
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Loops in AdS from conformal field theory
journal, July 2017

Aharony, Ofer; Alday, Luis F.; Bissi, Agnese
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Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?
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Romero-Bermúdez, Aurelio; Schalm, Koenraad; Scopelliti, Vincenzo
Journal of High Energy Physics, Vol. 2019, Issue 7
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The gravity dual of real-time CFT at finite temperature
journal, November 2018

Botta-Cantcheff, Marcelo; Martínez, Pedro J.; Silva, Guillermo A.
Journal of High Energy Physics, Vol. 2018, Issue 11
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Superluminal chaos after a quantum quench
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Balasubramanian, Vijay; Craps, Ben; De Clerck, Marine
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Quantum work statistics, Loschmidt echo and information scrambling
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Chenu, A.; Egusquiza, I. L.; Molina-Vilaplana, J.
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Optical radiation induced chaotic dynamics of electrons in a uniform magnetic field
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Yar, Abdullah; Zubair, Muhammad; Sabeeh, Kashif
Journal of Physics: Condensed Matter, Vol. 32, Issue 9
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Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise
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Tarkhov, Andrei E.; Fine, Boris V.
New Journal of Physics, Vol. 20, Issue 12
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Out-of-time-ordered correlation functions in open systems: A Feynman-Vernon influence functional approach
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Tuziemski, Jan
Physical Review A, Vol. 100, Issue 6
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Many-body systems with random spatially local interactions
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Morampudi, Siddhardh C.; Laumann, Chris R.
Physical Review B, Vol. 100, Issue 24
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Charge transport in graphene-based mesoscopic realizations of Sachdev-Ye-Kitaev models
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Can, Oguzhan; Nica, Emilian M.; Franz, Marcel
Physical Review B, Vol. 99, Issue 4
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Operator spreading in quantum maps
journal, March 2019

Moudgalya, Sanjay; Devakul, Trithep; von Keyserlingk, C. W.
Physical Review B, Vol. 99, Issue 9
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Holography for tensor models
journal, February 2020

de Mello Koch, Robert; Gossman, David; Tahiridimbisoa, Nirina Hasina
Physical Review D, Vol. 101, Issue 4
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Eigenstate thermalization hypothesis and out of time order correlators
journal, April 2019

Foini, Laura; Kurchan, Jorge
Physical Review E, Vol. 99, Issue 4
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Eigenstate Correlations, Thermalization, and the Butterfly Effect
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Chan, Amos; De Luca, Andrea; Chalker, J. T.
Physical Review Letters, Vol. 122, Issue 22
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Recent Developments in the Holographic Description of Quantum Chaos
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Jahnke, Viktor
Advances in High Energy Physics, Vol. 2019
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Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model.
text, January 2018

García-García, Antonio M.; Loureiro, Bruno; Romero-Bermúdez, Aurelio
Apollo - University of Cambridge Repository
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Tenfold way and many-body zero modes in the Sachdev-Ye-Kitaev model
text, January 2019

Behrends, Jan; Bardarson, Jh; Béri, B.
Apollo - University of Cambridge Repository
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Thickening and sickening the SYK model
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Khveshchenko, Dmitri
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Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators
text, January 2018

Rammensee, Josef; Urbina, Juan Diego; Richter, Klaus
Universität Regensburg
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Quasi-topological Ricci polynomial gravities
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Li, Yue-Zhou; Liu, Hai-Shan; Lü, H.
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Correlators on non-supersymmetric Wilson line in $$ \mathcal{N}=4 $$ SYM and AdS2/CFT1
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Beccaria, Matteo; Giombi, Simone; Tseytlin, Arkady A.
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Dynamics of quantum information
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Lewis-Swan, R. J.; Safavi-Naini, A.; Kaufman, A. M.
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Resistivity bound for hydrodynamic bad metals
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Lucas, Andrew; Hartnoll, Sean A.
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Scrambling and thermalization in a diffusive quantum many-body system
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Bohrdt, A.; Mendl, C. B.; Endres, M.
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Optimizing measurement strengths for qubit quasiprobabilities behind out-of-time-ordered correlators
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Mohseninia, Razieh; Alonso, José Raúl González; Dressel, Justin
Physical Review A, Vol. 100, Issue 6
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Scrambling of quantum information in quantum many-body systems
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Iyoda, Eiki; Sagawa, Takahiro
Physical Review A, Vol. 97, Issue 4
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Family of Sachdev-Ye-Kitaev models motivated by experimental considerations
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Lantagne-Hurtubise, Étienne; Li, Chengshu; Franz, Marcel
Physical Review B, Vol. 97, Issue 23
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Conformality of $1 / N$ corrections in Sachdev-Ye-Kitaev-like models
journal, December 2019

Dartois, Stéphane; Erbin, Harold; Mondal, Swapnamay
Physical Review D, Vol. 100, Issue 12
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Out of time ordered effective dynamics of a quartic oscillator
journal, January 2019

Chakrabarty, Bidisha; Chaudhuri, Soumyadeep
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Boomerang RG flows with intermediate conformal invariance
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Donos, Aristomenis; Gauntlett, Jerome P.; Rosen, Christopher
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Unscrambling the physics of out-of-time-order correlators
journal, October 2018

Swingle, Brian
Nature Physics, Vol. 14, Issue 10
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Thermalization and possible signatures of quantum chaos in complex crystalline materials
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Zhang, Jiecheng; Kountz, Erik D.; Behnia, Kamran
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Controlling chaos in the quantum regime using adaptive measurements
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Eastman, Jessica K.; Szigeti, Stuart S.; Hope, Joseph J.
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Global phase diagram of the one-dimensional Sachdev-Ye-Kitaev model at finite $N$
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Dai, Xin; Jian, Shao-Kai; Yao, Hong
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Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems
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Gopalakrishnan, Sarang; Huse, David A.; Khemani, Vedika
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Chaos and complexity in quantum mechanics
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Ali, Tibra; Bhattacharyya, Arpan; Haque, S. Shajidul
Physical Review D, Vol. 101, Issue 2
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Remarks on the Sachdev-Ye-Kitaev model
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Maldacena, Juan; Stanford, Douglas
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Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum
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Tomsovic, Steven; Lakshminarayan, Arul; Srivastava, Shashi C. L.
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Fluctuation Theorem for Many-Body Pure Quantum States
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Iyoda, Eiki; Kaneko, Kazuya; Sagawa, Takahiro
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On CFT and Quantum Chaos
text, January 2016

Turiaci, Gustavo; Verlinde, Herman
arXiv
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Slow scrambling in disordered quantum systems
text, January 2016

Swingle, Brian; Chowdhury, Debanjan
arXiv
DOI: 10.48550/arxiv.1608.03280

Thermodynamics of quantum information scrambling
text, January 2016

Campisi, Michele; Goold, John
arXiv
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Out-of-Time-Order Correlation in Marginal Many-Body Localized Systems
text, January 2016

Slagle, Kevin; Bi, Zhen; You, Yi-Zhuang
arXiv
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Black Holes and Random Matrices
text, January 2016

Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori
arXiv
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Hexagonalization of Correlation Functions
text, January 2016

Fleury, Thiago; Komatsu, Shota
arXiv
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Quantum bound to chaos and the semiclassical limit
preprint, January 2016

Kurchan, Jorge
arXiv
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Higher derivative corrections to incoherent metallic transport in holography
text, January 2016

Baggioli, Matteo; Goutéraux, Blaise; Kiritsis, Elias
arXiv
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Scale Anomalies, States, and Rates in Conformal Field Theory
text, January 2016

Gillioz, Marc; Lu, Xiaochuan; Luty, Markus A.
arXiv
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Entanglement, quantum randomness, and complexity beyond scrambling
text, January 2017

Liu, Zi-Wen; Lloyd, Seth; Zhu, Elton Yechao
arXiv
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Conformal Bootstrap in the Regge Limit
text, January 2017

Li, Daliang; Meltzer, David; Poland, David
arXiv
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Thermal diffusivity and chaos in metals without quasiparticles
text, January 2017

Blake, Mike; Davison, Richard A.; Sachdev, Subir
arXiv
DOI: 10.48550/arxiv.1705.07896

Operator Spreading in Random Unitary Circuits
text, January 2017

Nahum, Adam; Vijay, Sagar; Haah, Jeongwan
arXiv
DOI: 10.48550/arxiv.1705.08975

Relating out-of-time-order correlations to entanglement via multiple-quantum coherences
text, January 2017

Gärttner, Martin; Hauke, Philipp; Rey, Ana Maria
arXiv
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Nonexponential quantum decay under environmental decoherence
text, January 2017

Beau, M.; Kiukas, J.; Egusquiza, I. L.
arXiv
DOI: 10.48550/arxiv.1706.06943

Spectral Form Factor in Non-Gaussian Random Matrix Theories
text, January 2017

Gaikwad, Adwait; Sinha, Ritam
arXiv
DOI: 10.48550/arxiv.1706.07439

Teleportation Through the Wormhole
text, January 2017

Susskind, Leonard; Zhao, Ying
arXiv
DOI: 10.48550/arxiv.1707.04354

Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate
text, January 2017

Bianchi, Eugenio; Hackl, Lucas; Yokomizo, Nelson
arXiv
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Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks
text, January 2017

Kusuki, Yuya; Takayanagi, Tadashi
arXiv
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Complexity of the AdS Soliton
text, January 2017

Reynolds, Alan P.; Ross, Simon F.
arXiv
DOI: 10.48550/arxiv.1712.03732

Many-body localization, symmetry, and topology
text, January 2018

Parameswaran, S. A.; Vasseur, Romain
arXiv
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The non-Integrability of Strings in Massive Type IIA and their Holographic duals
text, January 2018

Nunez, Carlos; Penin, Jose Manuel; Roychowdhury, Dibakar
arXiv
DOI: 10.48550/arxiv.1802.04269

The Conformal Bootstrap at Finite Temperature
text, January 2018

Iliesiu, Luca; Koloğlu, Murat; Mahajan, Raghu
arXiv
DOI: 10.48550/arxiv.1802.10266

Entanglement Features of Random Hamiltonian Dynamics
text, January 2018

You, Yi-Zhuang; Gu, Yingfei
arXiv
DOI: 10.48550/arxiv.1803.10425

Falling Toward Charged Black Holes
text, January 2018

Brown, Adam R.; Gharibyan, Hrant; Streicher, Alexandre
arXiv
DOI: 10.48550/arxiv.1804.04156

Butterfly Velocity in Quadratic Gravity
text, January 2018

Huang, Wung-Hong
arXiv
DOI: 10.48550/arxiv.1804.05527

Light-ray operators in conformal field theory
text, January 2018

Kravchuk, Petr; Simmons-Duffin, David
arXiv
DOI: 10.48550/arxiv.1805.00098

Chaos and High Temperature Pure State Thermalization
text, January 2018

Lensky, Yuri D.; Qi, Xiao-Liang
arXiv
DOI: 10.48550/arxiv.1805.03675

Out-of-Time-Ordered-Correlator Quasiprobabilities Robustly Witness Scrambling
text, January 2018

Alonso, José Raúl González; Halpern, Nicole Yunger; Dressel, Justin
arXiv
DOI: 10.48550/arxiv.1806.09637

On Melonic Supertensor Models
text, January 2018

Chang, Chi-Ming; Colin-Ellerin, Sean; Rangamani, Mukund
arXiv
DOI: 10.48550/arxiv.1806.09903

Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators
text, January 2018

Liao, Yunxiang; Galitski, Victor
arXiv
DOI: 10.48550/arxiv.1807.09799

Rate of cluster decomposition via Fermat-Steiner point
text, January 2018

Avdoshkin, Alexander; Astrakhantsev, Lev; Dymarsky, Anatoly
arXiv
DOI: 10.48550/arxiv.1811.03633

Gapless odd-frequency superconductivity induced by the Sachdev-Ye-Kitaev model
text, January 2018

Gnezdilov, Nikolay V.
arXiv
DOI: 10.48550/arxiv.1812.00736

The Most Irrational Rational Theories
text, January 2018

Benjamin, Nathan; Dyer, Ethan; Fitzpatrick, A. Liam
arXiv
DOI: 10.48550/arxiv.1812.07579

Generalized hydrodynamics, quasiparticle diffusion, and anomalous local relaxation in random integrable spin chains
text, January 2019

Agrawal, Utkarsh; Gopalakrishnan, Sarang; Vasseur, Romain
arXiv
DOI: 10.48550/arxiv.1903.03122

Phases of scrambling in eigenstates
text, January 2019

Anous, Tarek; Sonner, Julian
arXiv
DOI: 10.48550/arxiv.1903.03143

Quantum information scrambling after a quantum quench
text, January 2019

Alba, Vincenzo; Calabrese, Pasquale
arXiv
DOI: 10.48550/arxiv.1903.09176

The chiral SYK model
text, January 2019

Lian, Biao; Sondhi, S. L.; Yang, Zhenbin
arXiv
DOI: 10.48550/arxiv.1906.03308

Dynamics of quantum information
text, January 2019

Lewis-Swan, R. J.; Safavi-Naini, A.; Kaufman, A. M.
arXiv
DOI: 10.48550/arxiv.1908.11747

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The Regge Limit for Green Functions in Conformal Field Theory text, January 2009

Spectral Gap and Exponential Decay of Correlations text, January 2005

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Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory journal, March 2019

Holographic excited states in AdS black holes journal, April 2019

The non-integrability of strings in massive type IIA and their holographic duals journal, June 2018

A crossing-symmetric OPE inversion formula journal, June 2019

Higher spin ANEC and the space of CFTs journal, July 2019

Quantum causal influence journal, July 2019

Quantum quenches and thermalization in SYK models journal, July 2019

An inelastic bound on chaos journal, July 2019

Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top journal, September 2019

Black holes as mirrors: quantum information in random subsystems
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Spectral Gap and Exponential Decay of Correlations
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Towards the fast scrambling conjecture
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Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics
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Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory
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Eikonal approximation in AdS/CFT: from shock waves to four-point functions
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Conformal bootstrap, universality and gravitational scattering
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An apologia for firewalls
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The black hole interpretation of string theory
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On gravitational shock waves in curved spacetimes
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Exact and approximate unitary 2-designs and their application to fidelity estimation
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The Regge limit for Green functions in conformal field theory
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Black holes and the butterfly effect
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The gravitational shock wave of a massless particle
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Fast scramblers
journal, October 2008

Stringy effects in scrambling
journal, May 2015

Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory
journal, January 2009

The finite group velocity of quantum spin systems
journal, September 1972

Comment on “Random Quantum Circuits are Approximate 2-designs” by A.W. Harrow and R.A. Low (Commun. Math. Phys. 291, 257–302 (2009))
journal, March 2011

Localized shocks
journal, March 2015

Causality constraints on corrections to the graviton three-point coupling
journal, February 2016

Efficiency of producing random unitary matrices with quantum circuits
journal, December 2008

Viscosity Bound and Causality Violation
journal, May 2008

The Pomeron and gauge/string duality
journal, December 2007

Holography from conformal field theory
journal, October 2009

Random Quantum Circuits are Approximate 2-designs
journal, July 2009

Convergence Rates for Arbitrary Statistical Moments of Random Quantum Circuits
journal, June 2010

Black holes as mirrors: quantum information in random subsystems
text, January 2007

Random Quantum Circuits are Approximate 2-designs
text, January 2008

The Regge Limit for Green Functions in Conformal Field Theory
text, January 2009

Spectral Gap and Exponential Decay of Correlations
text, January 2005

Out-of-time-ordered correlators in many-body localized systems: Out-of-time-ordered correlators in many-body localized systems
journal, December 2016

The most irrational rational theories
journal, April 2019

A bound on massive higher spin particles
journal, April 2019

Quantum Lyapunov spectrum
journal, April 2019

Solving the Schwarzian via the conformal bootstrap
journal, August 2017

Non-integrability and chaos with unquenched flavor
journal, October 2017

A species or weak-gravity bound for large N gauge theories coupled to gravity
journal, November 2019

Scattering effect on entanglement propagation in RCFTs
journal, December 2016

A note on the complex SYK model and warped CFTs
journal, December 2018

Nonuniqueness of Green’s functions at special points
journal, December 2019

Quantum Bound to Chaos and the Semiclassical Limit
journal, May 2018

Conservative chaos and invariant tori in the modified Sprott A system
journal, November 2019

The Lyapunov spectra of quantum thermalisation
journal, June 2019

Does gravity come from quantum information?
journal, October 2018

Verified quantum information scrambling
journal, March 2019

Entropic uncertainty relations for quantum information scrambling
journal, August 2019

Quantum chaos for the unitary Fermi gas from the generalized Boltzmann equations
journal, June 2019

Many-body localization, symmetry and topology
journal, July 2018

The Loschmidt-echo dynamics in a quantum chaos model
journal, March 2019

Nuclear magnetic resonance for quantum computing: Techniques and recent achievements
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Signature of quantum chaos in operator entanglement in 2d CFTs
journal, September 2019

An introduction to the SYK model
journal, July 2019

Scrambling in the Dicke model
journal, April 2019

Out-of-time-order correlation in marginal many-body localized systems
journal, April 2017

Gapless odd-frequency superconductivity induced by the Sachdev-Ye-Kitaev model
journal, January 2019

Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model
journal, April 2019

Teleportation through the wormhole
journal, August 2018

Complexity of Jackiw-Teitelboim gravity
journal, February 2019

Gauging classical and quantum integrability through out-of-time-ordered correlators
journal, October 2019

Semiclassical theory of out-of-time-order correlators for low-dimensional classically chaotic systems
journal, December 2018

Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices
journal, January 2019

Timescales in the quench dynamics of many-body quantum systems: Participation ratio versus out-of-time ordered correlator
journal, May 2019

Integrable Many-Body Quantum Floquet-Thouless Pumps
journal, October 2019

Out-of-time ordered correlators, complexity, and entropy in bipartite systems
journal, October 2019

Quantum out-of-equilibrium cosmology
journal, April 2019

Non‐Singular Black Holes Interiors Need Physics Beyond the Standard Model
journal, July 2019

Vacuum block thermalization in semi-classical 2d CFT
journal, February 2019

Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory
journal, March 2019

Holographic excited states in AdS black holes
journal, April 2019

The non-integrability of strings in massive type IIA and their holographic duals
journal, June 2018

A crossing-symmetric OPE inversion formula
journal, June 2019

Higher spin ANEC and the space of CFTs
journal, July 2019

Quantum causal influence
journal, July 2019

Quantum quenches and thermalization in SYK models
journal, July 2019

An inelastic bound on chaos
journal, July 2019

Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top
journal, September 2019

The many definitions of a black hole
journal, January 2019

Many-body localization and quantum thermalization
journal, October 2018

Accessing scrambling using matrix product operators
journal, November 2019