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.
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.
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.
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.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.
Out-of-time-ordered correlation functions (OTOCs) are presently being extensively debated as quantifiers of dynamical chaos in interacting quantum many-body systems. We argue that in quantum spin and fermionic systems, where all local operators are bounded, an OTOC of local observables is bounded as well and thus its exponential growth is merely transient. As a better measure of quantum chaos in such systems, we propose, and study, the density of the OTOC of extensive sums of local observables, which can exhibit indefinite growth in the thermodynamic limit. We demonstrate this for the kicked quantum Ising model by using large-scale numerical results and an analytic solution in the integrable regime. In a generic case, we observe the growth of the OTOC density to be linear in time. We prove that this density in general, locally interacting, nonintegrable quantum spin and fermionic dynamical systems exhibits growth that is at most polynomial in time---a phenomenon, which we term weak quantum chaos. In the special case of the model being integrable and the observables under consideration quadratic, the OTOC density saturates to a plateau.
We describe the dynamics of many-body quantum chaotic systems at all time scales by studying the Green's and out-of-time order correlation (OTOC) functions of the four-body, N-Majorana Sachdev-Ye-Kitaev model. By combining the scramblon formalism and random-matrix-theory techniques, we obtain analytical expressions for these functions at all times. The early exponential growth of the OTOC is followed by an exponential decay at a rate governed by that of the Green's function (the real part of the leading complex Ruelle-Pollicott resonances). For late times that scale exponentially with N, both functions have a dip-ramp-plateau pattern for Nmod8=2,6 that deviates substantially from the ergodic prediction due to local-in-energy correlations of matrix elements and eigenvalues, even after the Heisenberg time.
Abstract Out-of-time-ordered correlators (OTOCs), defined via the squared commutator of a time-evolving and a stationary operator, represent observables that provide useful indicators for chaos and the scrambling of information in complex quantum systems. Here we present a quasiclassical formalism of OTOCs, which is obtained from the semiclassical van Vleck–Gutzwiller propagator through the application of the diagonal approximation. For short evolution times, this quasiclassical approach yields the same result as the Wigner–Moyal formalism, i.e. OTOCs are classically described via the square of the Poisson bracket between the two involved observables, thus giving rise to an exponential growth in a chaotic regime. For long times, for which the semiclassical framework is, in principle, still valid, the diagonal approximation yields an asymptotic saturation value for the quasiclassical OTOC under the assumption of fully developed classical chaos. However, numerical simulations, carried out within chaotic few-site Bose–Hubbard systems in the absence and presence of periodic driving, demonstrate that this saturation value strongly underestimates the actual threshold value of the quantum OTOC, which is normally attained after the Ehrenfest time. This indicates that nondiagonal and hence genuinely quantum contributions, thus exceeding the framework of the quasiclassical description, are primarily responsible for describing OTOCs beyond the short-time regime.