System Symmetry and the Classification of Out-of-Time-Ordered Correlator Dynamics in Quantum Chaos

The symmetry of chaotic systems plays a pivotal role in determining the universality class of spectral statistics and dynamical behaviors, which can be described within the framework of random matrix theory. Understanding the influence of system symmetry on these behaviors is crucial for characterizing universal properties in quantum chaotic systems. In this work, we explore the universality of out-of-time-ordered correlator (OTOC) dynamics in quantum chaotic systems, focusing on the kicked rotor and the kicked Harper model. By modulating the periodically kicked potential, we control system symmetry to examine its impact on OTOC dynamics and level spacing distributions. Our results show that ensemble-averaged OTOC dynamics exhibit distinct universal behaviors depending on system symmetry, enabling classification through random matrix theory. These distinctions become evident after the localization time in localized regimes and emerge at specific time scales corresponding to the translational period of the Floquet operator in momentum space under quantum resonance conditions. Our findings provide a rigorous understanding of the relationship between symmetry and quantum chaotic dynamics, contributing to a deeper comprehension of universal behaviors in these systems.