Universal level statistics of the out-of-time-ordered operator

Abstract

The out-of-time-ordered correlator has been proposed as an indicator of chaos in quantum systems due to its simple interpretation in the semiclassical limit. In particular, its rate of possible exponential growth at → 0 is closely related to the classical Lyapunov exponent. Here we explore how this approach to quantum chaos relates to the random-matrix theoretical description. To do so, we introduce and study the level statistics of the logarithm of the out-of-time-ordered operator,Λ(t) = ln − [x(t),px(0)] 2 /(2t), that we dub the "Lyapunovian" or "Lyapunov operator" for brevity. The Lyapunovian's level statistics is calculated explicitly for the quantum stadium billiard. It is shown that in the bulk of the filtered spectrum, this statistics perfectly aligns with the Wigner-Dyson distribution. One of the advantages of looking at the spectral statistics of this operator is that it has a well-defined semiclassical limit where it reduces to the matrix of uncorrelated classical finite-time Lyapunov exponents in a partitioned phase space. We provide a heuristic picture interpolating these two limits using Moyal quantum mechanics. Our results show that the Lyapunov operator may serve as a useful tool to characterize quantum chaos and in particular quantum-toclassical correspondence in chaotic systems, by connecting the semiclassical Lyapunov growth at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.

Recent Developments in the Holographic Description of Quantum Chaos

“…In quantum field theories we can upgrade (11) to the case where the operators are separated in space Cpt, xq " x´rV p0, 0q, W pt, xqs 2 y β .…”

“…It was recently shown that the chaotic nature of many-body quantum systems can be diagnosed with certain out-of-time-order correlation (OTOC) functions which, in the gravitational description, are related to the collision of shock waves close to the black hole horizon [5][6][7][8][9]. In addition to being useful for diagnosing chaos in holographic systems and providing a deeper understanding for the inner-working mechanisms of gauge-gravity duality, OTOCs have also proved useful in characterizing chaos in more general non-holographic systems, including some simple models like the kicked-rotor [10], the stadium billiard [11], and the Dicke model [12].…”

Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

“…The more traditional approach to quantum chaos, however, focuses on the properties of the spectrum and uses level statistics as in random matrix theory (RMT) as its main signature [10][11][12][13]. There are several examples of cases where a correspondence between the exponential growth of the OTOC and level repulsion as in RMT has been found [14][15][16][17][18], but exceptions also exist [19][20][21]. In the present work, we propose a way to directly detect the effects of level repulsion in the evolution of a quantum system.…”

Timescales in the quench dynamics of many-body quantum systems: Participation ratio versus out-of-time ordered correlator

“…The quantum-classical correspondence between the exponential growth rate of the OTOC and the classical Lyapunov exponent has being numerically corroborated for finite systems with few degrees of freedom, such as one-body chaotic systems [27,28] and the Dicke model with two degrees of freedom [29]. However, little is known about this correspondence for finite quantum systems with many interacting particles.…”