Summary of our work
In our recently ArXived work, we studied the information scrambling in a chaotic bipartite system of kicked coupled tops with out-of-time ordered correlators (called OTOCs). We numerically investigated OTOCs for different types of initial operators, including the case of random operators where the operators are chosen randomly from the Gaussian unitary ensemble (GUE). In this study, we identified that the presence of a continuous symmetry such as conservation laws has a profound effect on the scrambling dynamics. Speaking of which, the presence of a conserved quantity implies different types of scrambling behaviors for various choices of initial operators depending on whether the operators commute with the symmetry operator.
The numerics confirm that the OTOCs exhibit power-law relaxation towards the equilibrium value in the limit of large Hilbert space dimension (i.e.in, the semi-classical limit). When the operators are randomly drawn from GUE, the averaged OTOC is related to the linear entanglement entropy of the Floquet operator. We also observe that the rate of short-time growth of the OTOCs, also known as the "quantum Lyapunov exponent," correlates remarkably well with the "classical Lyapunov exponents." Furthermore, we find signatures of chaos in the remarkable correspondence of the long-time average of OTOC calculated for the spin-coherent states and the classical phase space.
Significance of our results
Considerable efforts have been made to understand the dynamics of operator growth in generic many-body quantum systems with a single conserved quantity [V. Khemani et al., 2018 and C. von Keyserlingk et al., 2018]. Most of the works in this direction have been confined in generic many-body systems without obvious classical limits. Our work aims to fill the gap by studying the operator growth dynamics in the classical limit. You can find in our paper that most of the calculations have been done for huge angular momentum values. In many-body systems, the conservation law results in power-law decay of OTOCs towards equilibrium.
In our study of OTOCs, we found that in the classical limit, when the interaction is finite, the presence of a constant of motion (COM) slows down the scrambling of information. This result is not so obvious. While the power-law decay in the case of many-body systems with a COM stems from the extended system size in the real space and the locality of interactions, the power-law relaxation in our system seems to originate from the underlying classical dynamics and the phase space structure of the system.