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 can "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:1908.08955v1 [hep-th]

Slow scrambling in extremal BTZ and microstate geometries

“…However, this approximation clearly does not apply in the vacuum where there is no horizon. The applicability of the new approach summarized in section 2 to the zero temperature case was demonstrated in [29] where the quadratic growth associated to slow scrambling in vacuum was obtained, finding agreement with earlier CFT results [16]. Similarly, the results of [23] show that the shock wave diverges in the extremal limit when the approximation of placing it on the horizon is made.…”

“…A method to compute OTOCs within AdS/CFT, which is based on the geodesic approximation to the propagation of bulk fields in asymptotically AdS spacetimes, has been developed in [29]. We give a brief summary thereof in section 2.…”

“…All insertion points are also taken to lie asymptotically close to the spacetime conformal boundary in order to reproduce the OTOC of a dual conformal field theory. A method based on the geodesic approximation has been developed in a previous paper [29], which may be viewed as a position-space version of the one originally presented by Shenker and Stanford [15]. It was similarly constructed as the overlap…”

“…Within the geodesic approximation m V , m W 1 and the high-energy (eikonal) regime G N s 1, where s is the center-of-mass energy of the corresponding 2-to-2 scattering, the time-ordered amplitude (2.4) reduces to a simple phase e iδ which we describe below in more detail. This whole construction, originally presented in [29], is illustrated for the case of an extremal BTZ background geometry in figure 5. For -9 -JHEP03(2021)020…”

Holographic probes of inner horizons

“…The first term is the quadratic contribution of h to the Ricci scalar; there is no linear contribution since h solves the linearized equations of motion. The last two terms are in 47 It is necessary to use delta function identities such as δ (U ) = − δ(U ) U and U 2 δ(U ) 2 = 0. 48 These functions contain all angular dependence of the abstract shock profile α(φ) discussed in section 4.…”