Weak quantum chaos

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.

Figure 1

Density of the OTOC of extensive observables for one-dimensional KI model (7) with periodic boundary conditions: In (a) and (b), the magnetic field is transversal $\left(\phi =0\right)$ so the system is integrable (free), while in (c) and (d) the field is tilted $\left(\phi =\frac{\pi }{4}\right)$ so the model is nonintegrable. In (a) and (c), the observable is a sum of quadratic Majorana terms (10), while in (b) and (d) the observable is a sum of terms composed of infinite Majorana strings (composite). Here, $J=0.7,h=1.1$, but qualitatively similar behavior was found for other values of $J,h$. The numerically exact results for small system sizes are plotted with crosses. Results obtained with the numerical method based on typicality arguments (with a sample of $50\times 50$ random vectors) are plotted with error bars. The analytical solution for the integrable case and quadratic observable is plotted with a bold black line. The asymptotic behavior in the limits $N\to \infty$ and $t\to \infty$ is plotted with a dashed line. In (a), the dashed line was obtained analytically. In other cases, it is a numerical extrapolation. Numerical results start to deviate around $t\sim N/2$ due to finite size effects. The inset (i) shows the dependence of the plateau height on $J$ and $h$.