Superluminal chaos after a quantum quench

Abstract

Thermal states holographically dual to black holes in Einstein gravity display maximal Lyapunov growth as well as "butterfly effect cones". We study these effects in highly non-equilibrium states, obtained from an initial thermal state by the sudden injection of energy. We do this by computing out-of-time-order correlators (OTOCs) in BTZ-Vaidya spacetimes, which describe transitions between black holes at different temperatures. If both pairs of boundary operators appearing in the OTOC are inserted before the energy injection, we recover standard results, with butterfly effect cones displaying a light-cone structure. But when one pair of operators is inserted before and the other pair after the energy injection, the Lyapunov growth saturates the chaos bounds set by the local temperatures and the butterfly effect cones "open up", becoming superluminal, albeit in a way that does not violate causality. In the limiting case, in which the initial state is the vacuum, Lyapunov growth only starts after the energy injection. Our computations of the OTOCs are phrased in terms of gravitationally interacting particles, where fields are treated in a geodesic approximation and the eikonal phase shift is expressed in terms of stress tensors and shock waves associated to geodesics.

arXiv comment42 pages, 8 figures, corrected typos, matches published version

References in corpus (27)

• A bound on chaos
• Black holes and the butterfly effect
• Fast Scramblers
• Localized shocks
• Spectral Gap and Exponential Decay of Correlations
• Conformal Field Theory Interpretation of Black Hole Quasi-normal Modes
• Bulk vs. Boundary Dynamics in Anti-de Sitter Spacetime
• Multiple Shocks
• Two-dimensional conformal field theory and the butterfly effect
• The motion of point particles in curved spacetime
• Holographic Thermalization
• Holographic Particle Detection
• Eikonal Quantum Gravity and Planckian Scattering
• On Gravitational Shock Waves in Curved Spacetimes
• Eikonal Approximation in AdS/CFT: Resumming the Gravitational Loop Expansion
• Black Holes, Shock Waves, and Causality in the AdS/CFT Correspondence
• Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions
• Black Hole Collapse in the 1/c Expansion
• What Do CFTs Tell Us About Anti-de Sitter Spacetimes?
• Geodesic Structure of the 2 + 1 Black Hole
• Evolution of Two-Point Functions from Holography
• The Causal Structure of QED in Curved Spacetime: Analyticity and the Refractive Index
• Holographic probes of collapsing black holes
• van Vleck determinants: geodesic focussing and defocussing in Lorentzian spacetimes
• From Conformal Blocks to Path Integrals in the Vaidya Geometry
• Non-equilibrium scalar two point functions in AdS/CFT
• Back-Reaction and Complementarity in 1+1 Dilaton Gravity

Cited by in corpus (7)

• Chaos and complementarity in de Sitter space
• Complex geodesics in de Sitter space
• Slow scrambling in extremal BTZ and microstate geometries
• Soft graviton exchange and the information paradox
• Holographic Probes of Inner Horizons
• Reparametrization modes in 2d CFT and the effective theory of stress tensor exchanges
• Scrambling under quench