Superluminal chaos after a quantum quench

TL;DR

This paper investigates how chaos and butterfly effect cones behave in highly non-equilibrium states of holographic systems, revealing superluminal growth of chaos after a quantum quench without violating causality.

Contribution

It demonstrates superluminal butterfly effect cones in out-of-equilibrium holographic states, extending understanding of chaos bounds in quantum quenched systems.

Findings

Butterfly effect cones can become superluminal after a quantum quench.

Lyapunov growth saturates chaos bounds set by local temperatures.

Superluminal effects do not violate causality.

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.