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arXiv:1907.08030 · Journal of High Energy Physics

Scrambling in hyperbolic black holes: shock waves and pole-skipping

Yongjun Ahn, Viktor Jahnke, Hyun-Sik Jeong, Keun-Young Kim

Published Oct 1, 2019• 71 citations
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Abstract

A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result $$ {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} $$ v B T = 1 2 π ℓ = 1 d − 1 and the planar result $$ {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} $$ v B T ≫ 1 ℓ = d 2 d − 1 .

DOI
10.1007/jhep10(2019)257

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A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result vB(T=12πℓ)=1d−1{v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1}vB​(T=2πℓ1​)=d−11​ v B T = 1 2 π ℓ = 1 d − 1 and the planar result vB(T≫1ℓ)=d2(d−1){v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}}vB​(T≫ℓ1​)=2(d−1)d​​ v B T ≫ 1 ℓ = d 2 d − 1 .
A bstract Motivated by the recent connection between pole-skipping phenomena of two point functions and four point out-of-time-order correlators (OTOCs), we study the pole structure of thermal two-point functions in d -dimensional conformal field theories (CFTs) in hyperbolic space. We derive the pole-skipping points of two-point functions of scalar and vector fields by three methods (one field theoretic and two holographic methods) and confirm that they agree. We show that the leading pole-skipping point of two point functions is related with the late time behavior of conformal blocks and shadow conformal blocks in four-point OTOCs.
We describe a holographic mechanism for black hole formation via the collision of two shock waves in three-dimensional anti-de Sitter spacetime. In the dual conformal field theory (CFT), a two-shock-wave state corresponds to the insertion of two boosted precursor operators in complementary Rindler patches. Their operator product expansion is initially described by a universal mean-field spectrum of exchanged states, which is dominated by operator dimensions that grow exponentially in the boost parameter. We propose their mean value as diagnosing the mass of the collision product in the bulk. It crosses the CFT heavy state threshold after two scrambling times, in accordance with expectations about black hole formation in general relativity. Our analysis also allows us to identify the scrambling characteristics usually associated with out-of-time-order correlation functions, using only the internal composition of thermal in-time-order correlators.
The AdS/CFT correspondence and quantum entanglement (or quantum information). This is one of topics exciting many physicists in this decade. And black holes always play a role of inspiring lab in many cases. In this thesis, we will discuss about scrambling of quantum entanglement and complexity of states in AdS/CFT. In both topics, black holes are expected to be the fastest processors of quantum information. This thesis contains brief reviews of each topics and our own works. One work is an explicit calculation of fast scrambling time in two dimensional holographic CFTs and three dimensional BTZ black holes using the two-sided mutual information. This calculation is a first CFT counterpart of Shenker-Stanford’s work in the gravity dual and an extension of their shock-wave approximation to finite time region in the gravity side. Another work is about socalled optimization procedure of Euclidean path- integrals in CFTs. The optimization procedure gives us a new way to extract dual geometries only from CFT data. And it also gives a new proposal for the definition of complexity of states in CFTs.
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 TTT 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.
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.
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.