Weak Quantum Chaos

Abstract

Out-of-time-ordered correlation functions (OTOC's) 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, non-integrable 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.

Figures (3)

• Figure 1: Density of the OTOC of extensive observables for one-dimensional KI model \ref{['eq:HamiltonianKI']} with periodic boundary conditions is presented for four possible regimes. In the upper panels (A, B), the magnetic field is transversal ($\varphi=0$) so the system is integrable (free), while in the lower panels (C, D) the field is tilted ($\varphi=\frac{\pi}{4}$) so the model is non-integrable. In the left panels (A, C) the observable is a sum of quadratic Majorana terms \ref{['eq:MzMajorana']}, while in the right panels (B, D), the observable is a sum of terms composed of infinite Majorana strings (composite). Here $J=0.7$ and $h=1.1$ but the behaviour was found qualitatively similar for other values of $J,h$. The numerically exact results for small system sizes are plotted with crosses. Results obtained with numerical method based on typicality arguments (with a sample of $50\times50$ 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 behaviour in the limits $N\rightarrow\infty$ and $t\rightarrow\infty$ is plotted with a dashed line. In the integrable + quadratic case the dashed line is the result of our analytic solution. In other cases it is an extrapolation based on numerics. The 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 the parameters $J$ and $h$.
• Figure 2: The illustration of the main concepts needed in proving the polynomial upper bound on the dOTOC. At a given time, we can divide the $i$--$k$ plane into two regions and use different techniques to bound the contribution to the total bound on OTOC coming from each region. We will use the intuition implied by the LRT \ref{['eq:LiebRobinson1App']} that a commutator spreads essentially in causal-cone and is exponentially damped outside. The first region is the one where the causal-cones corresponding to the two commutators in \ref{['eq:ExtensiveOTOCapp']} overlap (for the particular choice of $i$ and $k$ in the drawing, this is the case for example at time $t'$). The leading order term in the bound on OTOC ($\propto t^3$) will come from this region. The second is the region where the light cones are well separated and can be embedded into semi-infinite intervals ($\Gamma_i$, $\Gamma_k$) with growing distance between them. This region will contribute subleading terms ($\propto t^2$).
• Figure 3: Floquet quasiparticle spectrum (eigenphases) of the kicked quantum Ising model \ref{['eq:EigenvalueSupp']}, \ref{['eq:MinusEigenvalueSupp']}. The full lines represents generic curves for the case of $J\neq h$, for which the spectrum has a gap. The dashed lines represent generic curves for the case of $J=h$, for which the gap closes and the system exhibits a Floquet analogue of a quantum phase transition.