Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model

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 ${\lambda }_L$. In particular, we find that ${\lambda }_L$ 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.

Figure 1

Comparison of TWA results to the exact dynamics. The top panel shows the time evolution of the occupation imbalance with $N=20$ modes, whereas in the bottom panel the individual mode occupation numbers are shown. In the bottom panel dashed lines correspond to the exact result. The inset shows the system size dependence of the time-averaged squared deviation of the TWA from the exact result.

Figure 2

Echo dynamics computed with TWA in comparison with exact results for $J\delta t=0.1$ and $N=16$ at quarter filling. The dashed line indicates the corresponding persistent echo peak height derived in Appendix pp1. The inset demonstrates that the normalized difference at the echo time is reduced as the system size is increased; the black line corresponds to an exponential fit.

Figure 3

TWA results for the divergence from the perfect echo computed for $N=20$ modes. As the perturbation strength $\delta t$ is decreased, the regime of exponential growth is extended, allowing for the identification of a Lyapunov exponent. The inset shows exact results for system sizes $N=8,12,16,20$ for $J\delta t=0.1$. These exact data are compatible with convergence towards the TWA result as $N\to \infty$.

Figure 4

(a) Echo divergence $\mathrm{\Delta }M\left(\tau \right)$ for different fixed $\tau$ as a function of the perturbation strength $\delta t$. Comparison of data obtained from exact full echo dynamics (dots) with the quadratic term in Eq. (3) alone (dashed lines). The double commutator is the single contribution to the echo divergence over a large range of $\tau$ and $\delta t$. (b) Corresponding TWA result. (c) Average divergence of initially close-by trajectories in phase space under TWA dynamics. A linear fit yields the estimate for the classical Lyapunov exponent.

Figure 5

Expansion dynamics from the uncorrelated initial state as observed in the single-mode occupation numbers $n_i\left(t\right)={\langle {\widehat{c}}_i^{†}{\widehat{c}}_i\rangle }_t$. The solid lines correspond to mean-field dynamics based on the most general equations of motion, Eq. (14). The dashed lines were obtained by computing the full quantum dynamics. The data shown are for $N=20$ at quarter filling.

Figure 6

Expansion dynamics from the uncorrelated initial state as observed in the single-mode occupation numbers $n_i\left(t\right)={\langle {\widehat{c}}_i^{†}{\widehat{c}}_i\rangle }_t$. The solid lines correspond to dynamics obtained based on the reduced mean-field equations of motion, Eq. (17), including fluctuations in the initial state by stochastic sampling of the initial conditions. The dashed lines were obtained by computing the full quantum dynamics. The data shown are for $N=20$ at quarter filling.

Figure 7

Full time evolution under imperfect effective time reversal as obtained by TWA in comparison with exact dynamics for different forward times $\tau$ with system size $N=20$ at quarter filling and $J\delta t=0.25$. The exact dynamics show a persistent echo signal, whereas the TWA echo vanishes at long forward times. The dashed line indicates the persistent peak height as given by Eq. (A2).

Figure 8

Finite-size analysis of exact results for the echo dynamics. The dashed lines indicate the saturation values obtained from the overlap of the perturbed with the unperturbed state. Here, $J\delta t=0.1$.

Figure 9

Finite-size analysis of results for the echo dynamics obtained using TWA.

Figure 10

Echo observed in the correlation average $\mathcal{C}$ [see Eq. (B1)] as a function of waiting time $\tau$. The exponential rate is the same as in the case of the occupation imbalance.