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arXiv:1911.04011

Out-of-time-order correlator in weakly perturbed integrable systems

Tomer Goldfriend, Jorge Kurchan B

Published Nov 10, 2019• 4 citations
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Abstract

Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time, but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent -- defined by the evolution of the 4-point out-of-time-order correlator (OTOC) -- of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. We find that {\em i)} in the semi-classical limit the quantum Lyapunov exponent is given by the classical one: it scales as $ε^{1/3}$, with $ε$ being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is $\sim ε^{-1}$). {\em ii)} in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and {\em iii)} for sufficiently small perturbations the $ε^{1/3}$ dependence is also suppressed -- another purely quantum effect which we explain. Several numerical examples which demonstrate the theoretical predictions are given. The implication for the results to the behavior of real near-integrable systems, and for quantum limits on chaos are briefly discussed.

DOI
10.48550/arxiv.1911.04011

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Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time, but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent -- defined by the evolution of the 4-point out-of-time-order correlator (OTOC) -- of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. We find that {\em i)} in the semi-classical limit the quantum Lyapunov exponent is given by the classical one: it scales as ε1/3ε^{1/3}ε1/3, with εεε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε−1\sim ε^{-1}∼ε−1). {\em ii)} in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and {\em iii)} for sufficiently small perturbations the ε1/3ε^{1/3}ε1/3 dependence is also suppressed -- another purely quantum effect which we explain. Several numerical examples which demonstrate the theoretical predictions are given. The implication for the results to the behavior of real near-integrable systems, and for quantum limits on chaos are briefly discussed.
Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time-the clearest example being the Solar System-but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent, defined by the evolution of the four-point out-of-time-order correlator (OTOC), of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. (1) We find that in the semiclassical limit the quantum Lyapunov exponent is given by the classical one: it scales as ε^{1/3}, with ε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε^{-1}). (2) We also find that in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and (3) for sufficiently small perturbations the ε^{1/3} dependence is also suppressed-another purely quantum effect which we explain. These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and kT/ℏ may stay finite at a low temperature T.
It has been a long-standing challenge to expand the notion of exponential sensitivity to small perturbations in the initial conditions from the realm of classical chaos to that of many-body quantum systems. Recently, exponential growth in a measure known as the out-of-time-order commutator (OTOC) has been proposed as a diagnostic of chaos in the setting of many-body quantum systems. The authors examine the behavior of the OTOC along rays of different velocities for a variety of spatially local extended many-body quantum systems, both integrable and nonintegrable. They find that the velocity-dependent Lyapunov exponents are negative for velocities greater than a characteristic ``butterfly speed'', which defines the light cone for the spreading of operators. It is demonstrated that a regime with well-defined positive Lyapunov exponents inside the light-cone may only exist for classical, semiclassical, weakly interacting, or large-NNN systems, but not for fully quantum systems with strong short-range interactions and local Hilbert space dimensions of order one.
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because in the semi-classical limit, ℏ→0\hbar \to 0ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator, C(t)C(t)C(t), for the classical and quantum kicked rotor -- a textbook driven chaotic system -- and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are in general distinct quantities, corresponding to the logarithm of phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength, where no classical chaos exists globally. In this case, the Lyapunov exponent vanishes, while the OTOC's growth rate may remain finite corresponding to disconnected local chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time tEt_EtE​: transitioning from a time-independent value of t−1ln⁡C(t)t^{-1} \ln{C(t)}t−1lnC(t) at ttEt t_EttE​. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator C(t) for the classical and quantum kicked rotor-a textbook driven chaotic system-and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K→0, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_{E}: transitioning from a time-independent value of t^{-1}lnC(t) at t<t_{E} to its monotonic decrease with time at t>t_{E}. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996)PRBMDO0163-182910.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
We examine three billiard systems -- the cardioid, diamond (Superman), and Sinai billiards -- all of which are known to be classically chaotic. We compute their classical Lyapunov exponents, and using out-of-time-order correlators (OTOCs) in the semi-classical regime, we also derive their quantum Lyapunov exponents. We observe that the classical and quantum Lyapunov exponents are in agreement, strengthening the role of OTOCs as a diagnostic for quantum chaos in billiard systems. At very low temperatures, the OTOC of the Sinai billiard shows sharp growth, a phenomenon absent in the other two billiards. We identify the source of this anomalous behaviour in the geometry of the ground state wave function of the Sinai billiard, which is more sensitive to the curvature of the billiard compared to the other billiards. We also remark on the late-time behaviour of the OTOCs and how the scrambling/Ehrenfest time is related to the temperature of quantum billiards.
The out-of-time-order correlator (OTOC) is studied for a bosonic quantum lattice model. We gain its classical analog through the replacement of both commutators appearing in the quantum correlator by a corresponding Poisson bracket. The evaluation of the Poisson bracket is then performed in a complex-valued description of the Hamiltonian dynamics and, for the initial choice of quantum operators to be site-specific annihilators, turns out to be given by the expectation value of the absolute square of a specific element of the complex-valued monodromy matrix. The growth rate of this expectation value is compared to a typical chaos indicator, the mean finite-time Lyapunov exponent (FTLE). In both cases the numerical phase-space average is weighted by a Wigner function corresponding to a multimode coherent state. For a three-well Bose-Hubbard model in the Mott insulator regime, it is found that although, on the level of single trajectories, FTLE and classical OTOC show similar long-time behavior, after averaging, they exhibit a marked difference [which for purely chaotic initial conditions is close to ln(sqrt[2])], rooting in the different order the logarithm and the average are taken. This observation is an example of the relevance of the fluctuations of the FTLE to correctly explain the quantitative difference between OTOC growth rate and FTLE in a prototypical many-body system.