
Quantum Chaos and Black Holes:
the Lyapunov Exponent and Butterfly Velocity
Viktor Jahnke, KeunYoung Kim, and Mitsuhiro Nishida
School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju, Korea
BRIEF REVIEW OF QUANTUM CHAOS
While classical chaos is relatively well understood, the characterization of quantum chaos is fairly complicated. Possible approaches include, for instance, the eigenstate thermalization hypothesis (ETH) [1], level spacing statistics described by the random matrix theory (RMT) [2], and outoftime order correlators (OTOCs) [3]. Despite the insights provided by these approaches, a more satisfactory understanding of quantum chaos remains elusive. Here we briefly review the concepts of ETH and level spacing statistics.
Eigenstate Thermalization Hypothesis
The ETH seeks to explain how some isolated quantum systems thermalize, i.e., how an initially farfromequilibrium state evolves to a state that appears to be in thermal equilibrium.
ETH proposes that the energy eigenstates of chaotic manybody quantum systems are indistinguishable from a thermal state when probed by local operators [1].
Level Spacing Statistics
Another feature of quantum chaotic systems is level spacing statistics described by RMT [2]. This basically states that the (properly normalized) spacing between consecutive energy eigenvalues obeys a WignerDyson distribution, which also describes the level spacing statistics of random matrices. By contrast, in integrable systems, the level spacing statistics is usually described by a Poisson distribution, but there are exceptions.
OTOCS AS A DIAGNOSIS OF QUANTUM CHAOS
General definition
First, let us explain OTOCs in quantum mechanics. We would like to measure the chaotic effect between two operators W(t) and V(0) by computing a thermal expectation value of the square of a commutator. This expectation value can be expanded into four pieces:
 (1)

The last two terms are fourpoint OTOCs in quantum mechanics.
Next, consider OTOCs <W(t,x)V(0)W(t,x)V(0)> in quantum field theories (QFTs), where t and x are time and space coordinates of W(t,x). OTOCs in QFTs can be computed by performing an analytic continuation of Euclidean correlation functions. In this analytic continuation, the order of Euclidean time in the original Euclidean correlation functions is important for out of time order in real time.
In some chaotic systems, the OTOCs at a certain time t behave as
 (2)

with positive λ. Now, λ is the quantum version of the Lyapunov exponent, which characterizes the divergence of initially nearby trajectories in classical chaotic motion; and v is the butterfly velocity, which describes the speed at which information propagates through space.
QFT examples
We give examples of QFTs where we can compute OTOCs. In the below examples, the Lyapunov exponent saturates the upper bound [4], and it is a characteristic property of QFTs that have the gravity duals.
The first example is the SYK model [5]. The SYK model is a quantum mechanical system with N Majorana fermions. These fermions interact with random couplings from a Gaussian distribution. In the large N and strong coupling limit, we obtain the upper bound value of the Lyapunov exponent.
The second example is seen in twodimensional holographic conformal field theories [6]. The computation of OTOCs from the Virasoro identity block with a large central charge reproduces the holographic computation of OTOCs.
Holographic calculation of OTOCs
In the context of the gaugegravity duality [3], OTOCs are related to a high energy shock wave collision that takes place close to a black hole horizon. The collision is dominated by the gravitational interaction, which leads to the universal chaotic behavior of the boundary theory OTOCs with λ = 2𝜋T, where T is the Hawking temperature.
Holographic calculation shows that black holes saturate the chaos bound, i.e., λ ≤ 2𝜋T. This singles out holographic systems as being maximally chaotic. This special property of holographic systems helped in the construction of simple models for holography, e.g., SYK model / JT gravity [5].
The Lyapunov exponent λ and the butterfly velocity (v) fully characterize the universal part of the OTOC and are important parameters characterizing the chaotic behavior.
Chaos bound in rotating black holes
A large class of black holes is characterized by a maximal Lyapunov exponent, namely, λ = 2𝜋T [4]. In fact, it is actually very difficult to find examples of black holes that go beyond the maximal exponent; the only known example of such a black hole in which the Lyapunov exponent does not take such a simple form is provided by the rotating BTZ black holes. In this case, the angular momentum affects the Lyapunov exponent, leading to contributions that either fail to saturate the chaos bound, or violate it [7].
POLESKIPPING PHENOMENON
The parameters λ and v can also be extracted from momentumspace energydensity retarded twopoint functions G(ω,k). Here, ω is the frequency and k the spatial momentum.
By writing G(ω,k) = b(ω,k)/a(ω,k), one can find the poles and the zeros of G(ω,k) as the zeros of a(ω,k) and b(ω,k) respectively. There are, however, some special points (ω*,k*) at which a(ω*,k*) = b(ω*,k*)=0. Those are called poleskipping points, and they were shown to be related to λ and v [8]:
 (3)

Since G(ω,k) characterizes the transport properties of the theory, the poleskipping phenomenon points to an interesting connection between chaos and hydrodynamics.
Poleskipping in hyperbolic black holes
The poleskipping phenomenon was first observed/studied for planar black holes, but it was later realized that the poleskipping phenomenon also happens for hyperbolic black holes [9]. In that case, the momentumspace retarded energydensity twopoint function G(ω,L) is characterized by the frequency ω and by an angularmomentumlike quantum number L.
The poleskipping points (ω*,k*) of G(ω,L) were shown to be related to the Lyapunov exponent and butterfly velocity extracted from OTOCs.
Poleskipping for scalar and vector fields
The poleskipping phenomenon also occurs for other fields rather than the gravitational sound mode that characterizes fluctuations in energy density. In fact, it was shown that it also occurs for scalar and vector fields, and also for other sectors of gravitational perturbations [10].
In [10], the authors considered a (d+1)dimensional RindlerAdS geometry and studied the poleskipping points of scalar and vector fields using several different methods, including both CFT and holographic calculations. They observed a precise matching between the poleskipping point obtained in both sides of the gaugegravity duality. Moreover, they showed that poleskipping points are related to the latetime behavior of conformal blocks and shadow conformal blocks.
References
[1] M. Srednicki, "The approach to thermal equilibrium in quantized chaotic systems", J. Phys. A 32, 1163 (1999), arXiv:condmat/9809360.
[2] M. L. Mehta, Random matrices, vol. 142. Academic press, 2004.
[3] S. H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132, [arXiv:1412.6087].
[4] J. Maldacena, S. H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106, [1503.01409].
[5] A. Kitaev, A simple model of quantum holography, (2015) http://online.kitp.ucsb.edu/online/entangled15/kitaev/, http://online.kitp.ucsb.edu/online/entangled15/kitaev2//, Talks at KITP, April 7, 2015 and May 27, (2015).
[6] D. A. Roberts and D. Stanford, Twodimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603, [1412.5123].
[7] V. Jahnke, K.Y. Kim, and J. Yoon, On the Chaos Bound in Rotating Black Holes, JHEP 05 (2019) 037, [arXiv:1903.09086].
[8] S. Grozdanov, K. Schalm, and V. Scopelliti, Black hole scrambling from hydrodynamics, Phys. Rev. Lett. 120 (2018), no. 23 231601, [arXiv:1710.00921].
[9] Y. Ahn, V. Jahnke, H.S. Jeong and K.Y. Kim, Scrambling in Hyperbolic Black Holes: shock waves and poleskipping, JHEP 10 (2019) 257, [1907.08030].
[10] Yongjun Ahn, Viktor Jahnke, HyunSik Jeong, KeunYoung Kim, KyungSun Lee, and Mitsuhiro Nishida, Poleskipping of scalar and vector fields in hyperbolic space: conformal blocks and holography arXiv:2006.00974.

Viktor Jahnke is a postdoctoral researcher in the Department of Physics and Photon Science at the Gwangju Institute of Science and Technology (GIST), Korea. After receiving his PhD from the University of Sao Paulo, Brazil, he worked at Universidad Nacional Autonoma de Mexico before joining GIST in 2018. His research field is theoretical physics. 

KeunYoung Kim is the executive editor of Journal of the Korean Physical Society, and a professor in the Department of Physics and Photon Science at the Gwangju Institute of Science and Technology (GIST), Korea. After receiving his PhD from the State University of New York at Stony Brook, USA, he worked at Southampton University, UK, and the University of Amsterdam, Netherlands, before joining GIST in 2013. His research field is theoretical physics. 

Mitsuhiro Nishida is a postdoctoral researcher in the Department of Physics and Photon Science at the Gwangju Institute of Science and Technology (GIST), Korea. He received his PhD from the Department of Physics, Osaka University, Japan in 2017. He is interested in research of the gaugegravity duality. 
