Chaos in the butterfly cone

Abstract

A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent, $\lambda (\mathbf{v})$. We show that this exponent is bounded inside the butterfly cone as $\lambda (\mathbf{v})\le 2\pi T(1-|\mathbf{v}|/v_B)$, where $T$ is the temperature and $v_B$ is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study $\lambda (\mathbf{v})$ in some examples such as two dimensional SYK models and holographic gauge theories, and observe that in these systems the bound gets saturated at some critical velocity $v_{\ast }<v_B$. In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where $\lambda (\mathbf{v})$ is related to the spin of the leading large $N$ Regge trajectory, and controls the four point function in an interpolating regime between the Regge and the light cone limit. Finally, we comment on the generalization of the chaos bound to boosted and rotating ensembles and clarify some recent results on this in the literature.

arXiv comment35 pages, 15 figures. v2: published version

References in corpus (51)

• A bound on chaos
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Comments on the Sachdev-Ye-Kitaev model
• Black holes and the butterfly effect
• The Spectrum in the Sachdev-Ye-Kitaev Model
• Localized shocks
• Bekenstein-Hawking Entropy and Strange Metals
• The Pomeron and Gauge/String Duality
• Analyticity in Spin in Conformal Theories
• Two-dimensional conformal field theory and the butterfly effect
• Convexity and Liberation at Large Spin
• Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models
• Conformal Regge theory
• A strongly correlated metal built from Sachdev-Ye-Kitaev models
• Light-ray operators in conformal field theory
• Causality Constraints in Conformal Field Theory
• Averaged Null Energy Condition from Causality
• Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches
• Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves
• AdS${}_2$ holography and the SYK model
• On entanglement spreading in chaotic systems
• Many-body chaos at weak coupling
• More on Supersymmetric and 2d Analogs of the SYK Model
• Black Holes, Shock Waves, and Causality in the AdS/CFT Correspondence
• Velocity-dependent Lyapunov exponents in many-body quantum, semi-classical, and classical chaos
• Bounding the Space of Holographic CFTs with Chaos
• Onset of many-body chaos in the $O(N)$ model
• Bounding wide composite vector resonances at the LHC
• Bounds for OPE coefficients on the Regge trajectory
• Eikonal Methods in AdS/CFT: Regge Theory and Multi-Reggeon Exchange
• On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections
• On the relation between the magnitude and exponent of OTOCs
• Scrambling in Hyperbolic Black Holes: shock waves and pole-skipping
• A perturbative perspective on self-supporting wormholes
• On entanglement spreading from holography
• On the Chaos Bound in Rotating Black Holes
• Transport and chaos in lattice Sachdev-Ye-Kitaev models
• Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators
• An exact slope for AdS/CFT
• A conformal block Farey tail
• Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?
• Strong Coupling Expansion for the Conformal Pomeron/Odderon Trajectories
• BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin
• Butterflies with rotation and charge
• Thermalization and chaos in QED${}_3$
• $\mathcal{N}=(0,2)$ SYK, Chaos and Higher-Spins
• The chiral SYK model
• Scrambling time from local perturbations of the rotating BTZ black hole
• On the Derivation of the Exact Slope Function
• Chaos in the Fishnet
• Global Symmetry and Maximal Chaos

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• Absence of fast scrambling in thermodynamically stable long-range interacting systems
• Quantum Chaos in Topologically Massive Gravity
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• Pole skipping in holographic theories with bosonic fields
• Chaos and pole skipping in CFT${}_2$
• Phase transition and chaos in charged SYK model
• Pole skipping away from maximal chaos
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• Dirac Fast Scramblers
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• Fast Scrambling of Mutual Information in Kerr-AdS${}_4$
• Fast Scrambling of mutual information in Kerr-AdS${}_{\text{5}}$
• Localized chaos due to rotating shock waves in Kerr-AdS black holes and their ultraspinning version
• Collisions of localized shocks and quantum circuits
• Chiral Sachdev-Ye model: Integrability and chaos of anyons in 1+1d
• Global Symmetry and Maximal Chaos
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• Holographic study of shear viscosity and butterfly velocity for magnetic field-driven quantum criticality
• Jackiw-Teitelboim gravity and near-extremal BTZ thermodynamics
• JT gravity and near-extremal thermodynamics for Kerr black holes in $Ad{S}_{4,5}$ for rotating perturbations
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• Holographic Black Hole Formation and Scrambling in Time-Ordered Correlators
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