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arXiv:1802.06796 · Physical review. B./Physical review. B

Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model

Markus Schmitt, Dries Sels, Stefan K. Kehrein, Anatoli Polkovnikov

Published Apr 8, 2019• 45 citations
arXivHTMLPDF

Abstract

The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question has gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach using the truncated Wigner approximation, which accounts for nonvanishing quantum fluctuations that are essential for the dynamics. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows us to identify a Lyapunov exponent ${\ensuremath{\lambda}}_{L}$. In particular, we find that ${\ensuremath{\lambda}}_{L}$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.

DOI
10.1103/physrevb.99.134301

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The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question has gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach using the truncated Wigner approximation, which accounts for nonvanishing quantum fluctuations that are essential for the dynamics. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows us to identify a Lyapunov exponent \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​. In particular, we find that \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.
Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-N) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins N. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on N and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.
A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large- q limit of a q -body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q ≥ 4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024
The Loschmidt echo (LE) is a magnitude that measures the sensitivity of quantum dynamics to perturbations in the Hamiltonian. For a certain regime of the parameters, the LE decays exponentially with a rate given by the Lyapunov exponent of the underlying classically chaotic system. We develop a semiclassical theory, supported by numerical results in a Lorentz gas model, which allows us to establish and characterize the universality of this Lyapunov regime. In particular, the universality is evidenced by the semiclassical limit of the de Broglie wavelength going to zero, the behavior for times longer than Ehrenfest time, the insensitivity with respect to the form of the perturbation, and the behavior of individual (nonaveraged) initial conditions. Finally, by elaborating a semiclassical approximation to the Wigner function, we are able to distinguish between classical and quantum origin for the different terms of the LE. This approach renders an understanding for the persistence of the Lyapunov regime after the Ehrenfest time, as well as a reinterpretation of our results in terms of the quantum-classical transition.
A method has been proposed to calculate the out-of-time time ordered correlator in the generalization of the Sachdev–Ye–Kitaev model with a nonzero spatial dimension. The result is applicable not only at small times, when the chaotic properties of the system are developed weakly, but also at large times of about the Erenfest time. It has been shown that information on the applied perturbation, which is described by this correlator, propagates ballistically in the form of a front. The velocity of the front for models of this type has been calculated for the first time.
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.
We suggest a new indicator of quantum chaos based on the logarithmic out-of-time-order correlator. On the one hand, this indicator correctly reproduces the average classical Lyapunov exponent in the semiclassical limit and directly links the definitions of quantum chaos and classical K system. On the other hand, it can be analytically calculated using the replica trick and the Schwinger-Keldysh diagram technique on a 2n2n2n-fold Keldysh contour. To illustrate this approach, we consider several one-dimensional systems, including the quantum cat map, and three paradigmatic large-NNN models, including the Sachdev-Ye-Kitaev model. Furthermore, we find that correlations between replicas can reduce the magnitude of the Lyapunov exponent compared to estimates based on conventional out-of-time-order correlators.