Abstract Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display “slow scrambling”, characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory. Next we study the extent to which these OTOCs are modified in certain “superstrata”, horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.
Abstract We compute out-of-time-order correlators (OTOCs) in two-dimensional holographic conformal field theories (CFTs) with different left- and right-moving temperatures. Depending on whether the CFT lives on a spatial line or circle, the dual bulk geometry is a boosted BTZ black brane or a rotating BTZ black hole. In the case when the spatial direction is non-compact, we generalise a computation of Roberts and Stanford and show that to reproduce the correct bulk answer a maximal channel contribution needs to be selected when using the identity block approximation. We use the correspondence between global conformal blocks and geodesic Witten diagrams to extend our results to CFTs on a spatial circle. In [1] it was shown that the OTOC for a rotating BTZ black hole exhibits a periodic modulation about an average exponential decay with Lyapunov exponent 2π/β. In the extremal limit where the black hole is maximally rotating, it was shown in [2] that the OTOC exhibits an average cubic growth, on which is superposed a sawtooth pattern which has small periods of Lyapunov growth due to the non-zero temperature of left-movers in the dual CFT. Our computations explain these results from a dual CFT perspective.
We study from the perspective of quantum information scrambling an acoustic black hole modeled by two semi-infinite, stationary, one-dimensional condensates, connected by a spatial step-like discontinuity, and flowing, respectively, at subsonic and supersonic velocities. We develop a simple analytical treatment based on the Bogoliubov theory of quantum fluctuations which is sufficient to derive an analogue Hawking emission, and we compute out-of-time order correlations (OTOCs) of the Bose density field. We find that a large class of sonic black holes presents slow scrambling contrary to their astrophysical counterparts. This manifests itself in a power law growth of OTOCs in contrast to the exponential increase in time expected for fast scramblers.
We calculate the four-wave scattering amplitude in the background of an AdS traversable wormhole in 2+1 dimensions created by a nonlocal coupling of AdS boundaries in the BTZ black hole background. The holographic dual of this setup is a pair of CFTs coupled via a double-trace deformation, the scattering amplitude giving the out-of-time ordered correlator (OTOC) in CFT. Short-living wormholes always exhibit a regime of fast scrambling, saturating the MSS bound for the Lyapunov exponent but in the early-time regime the scrambling can be slower, with a Lyapunov exponent linear in
T but below the MSS bound. For long-living (near-eternal) wormholes however the numerics suggests the existence of another regime, with a drastic (exponential) slowdown of scrambling and an exponentially small Lyapunov exponent. Our findings have parallels in the SYK model, and may indicate certain limitations of wormhole teleportation protocols previously studied in the literature.
A bstract We explore the physics of scrambling in the moving mirror models, in which a two-dimensional CFT is subjected to a time-dependent boundary condition. It is well-known that by choosing an appropriate mirror profile, one can model quantum aspects of black holes in two dimensions, ranging from Hawking radiation in an eternal black hole (for an “escaping mirror”) to the recent realization of Page curve in evaporating black holes (for a “kink mirror”). We explore a class of OTOCs in the presence of such a boundary and explicitly demonstrate the following primary aspects: First, we show that the dynamical CFT data directly affect an OTOC and maximally chaotic scrambling occurs for the escaping mirror for a large- c CFT with identity block dominance. We further show that the exponential growth of OTOC associated with the physics of scrambling yields a power-law growth in the model for evaporating black holes which demonstrates unitary dynamics in terms of a Page curve. We also demonstrate that, by tuning a parameter, one can naturally interpolate between an exponential growth associated with scrambling and a power-law growth in unitary dynamics. Our work explicitly exhibits the role of higher-point functions in CFT dynamics as well as the distinction between scrambling and Page curve. We also discuss several future possibilities based on this class of models.
Abstract We introduce a local operational protocol for measuring quantum information scrambling in a curved spacetime background. In this framework, two Unruh-DeWitt (UDW) detectors, coupled to a massless conformally coupled scalar field in the Hartle–Hawking vacuum of the non-rotating ( 2 + 1 ) -dimensional Banados–Teitelboim–Zanelli (BTZ) black hole, serve as local probes. Evaluating the detector out-of-time-ordered correlator requires a fourth-order Dyson series expansion. Within the scrambling regime, this correlator exhibits exponential growth with Lyapunov exponent, exactly saturating the Maldacena–Shenker–Stanford bound at the global Hawking temperature. Governed by the black hole surface gravity, the Lyapunov exponent is universal and independent of detector positions, energy gaps, and coupling strengths. Furthermore, the scrambling amplitude shows a monotonic dependence on the radial positions of the detectors, vanishing at the event horizon and increasing outward. Local two-level probes thus witness the maximal chaos of the BTZ geometry without reconstructing the global quantum field state.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate
Λ as a function of the number of degrees of freedom,
f, and inverse temperature,
β.
We show that the scaled growth rate,
Λ/f, can be described by a universal function of
fβ and displays a cross-over from classical to quantum behavior as we increase
f and/or lower the temperature. In the deep quantum regime of infinite
f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.