Diagnosing chaos in a periodically driven Ising model with a ramping field via out-of-time-order correlation saturation

The dynamic region of out-of-time-ordered correlators (OTOCs) serves as a powerful indicator of chaos in classical and semiclassical systems, capturing the characteristic exponential growth. However, in spin systems, the dynamic region of OTOCs does not reliably quantify chaos as both integrable and chaotic systems display similar behavior. Instead, we utilize the saturation behavior of OTOCs to differentiate between chaotic and integrable regimes. In integrable systems, the saturation region of OTOCs exhibits oscillatory behavior, while in chaotic systems, it shows a stable saturation. To evaluate this distinction, we investigate a time-dependent Ising spin system subjected to a linearly ramping transverse field, analyzing both integrable (without longitudinal field) and non-integrable (with longitudinal field) scenarios. This setup accelerates system ergodicity due to the introduction of an additional time scale within the system, enhancing chaotic dynamics in the non-integrable regime, and provides a compelling model for studying the interplay between integrability and chaos in quantum systems. To further support our findings, we investigate the level spacing distribution of time-dependent unitary operators, which effectively distinguishes chaotic from regular regions in our system and corroborates the results obtained from the saturation behavior of the OTOC. Additionally, we calculate a metric for the normalized Fourier spectrum of the OTOC which is dependent on the number of frequency components present to gain insights into the observed oscillations and its dependence on the ramping field.