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arXiv:2411.16274 · Journal of Physics A Mathematical and Theoretical

Out-of-time ordered correlator for a chaotic many-body quantum system

Hans A. Weidenmüller

Published Apr 28, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract Using the parametric representation of a chaotic many-body quantum system derived earlier, we calculate explicitly the large-time dependence and asymptotic value of the out-of-time correlator (OTOC) of that system. The dependence on time t is determined by Δ t / ℏ . Here Δ is the energy correlation width within which the Bohigas–Giannoni–Schmit conjecture applies. We conjecture that Δ is universally related to the leading Ljapunov coefficient of the corresponding classical system by Δ = ℏ λ max . Then the large-time behavior of OTOC is given by the dimensionless parameter λ max t .

DOI
10.1088/1751-8121/add198

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Abstract Using the parametric representation of a chaotic many-body quantum system derived earlier, we calculate explicitly the large-time dependence and asymptotic value of the out-of-time correlator (OTOC) of that system. The dependence on time t is determined by Δ t / ℏ . Here Δ is the energy correlation width within which the Bohigas–Giannoni–Schmit conjecture applies. We conjecture that Δ is universally related to the leading Ljapunov coefficient of the corresponding classical system by Δ = ℏ λ max . Then the large-time behavior of OTOC is given by the dimensionless parameter λ max t .
Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τ_{E} in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τ_{E}=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τ_{E}, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems.
Abstract We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.
In this work, we show, that relative oscillations in asymptotic times of microcanonical Out-of-Time-Order correlators (OTOCs) are suppressed by chaos with the size of the sys- tem. Therefore they can serve as an indicator of quantum chaos in quantum systems. We demonstrate this phenomenon on numerical results of the Out-of-Time-Order correlators from a model with hamiltonian based on the u(3) algebra. Firstly we prove, that this model can be used not only for the description of molecular vibrations, but also Bose- Einstein condensate. Then we employ several methods for the analysis of both quantum and classical chaos, namely spectral statistics such as Nearest Neighbour Spacing Distri- bution and Brody distribution, Inverse Participation Ratio, Poincaré sections, Lyapunov exponent, and the chaotic fraction of classical phase space. We compare these results with relative oscillations of microcanonical Out-of-Time-Order correlators and show, that their suppression corresponds with the chaoticity of the quantum system and the fraction of chaotic regions of the classical phase space. 1
Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.
The time-dependence of multipoint observable correlation functions are essential quantities in analysis and simulation of quantum dynamics. Open quantum systems approaches utilize two-point correlations to describe the influence of an environment on a system of interest, and in studies of chaotic quantum system, the out-of-time-ordered correlator (OTOC) is used to probe chaoticity of dynamics. In this work we analytically derive the time dependence of multipoint observable correlation functions in quantum systems from a random matrix theoretic approach, with the highest-order function of interest being the OTOC. We find in each case that dynamical contributions are related to a simple function, related to the Fourier transform of coarse-grained wave functions. We compare the predicted dynamics to exact numerical experiments in a spin chain for various physical observables. We comment on implications toward the emergence of Markovianity and quantum regression in closed quantum systems, as well as relate our results to known bounds on chaotic dynamics.