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

Abstract

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. In contrast, this signature fails to appear in spin systems, where even chaotic dynamics lack such exponential escalation, making this region an unreliable marker of chaos. To address this limitation, we turn to 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. The ramping introduces a time-dependent increase of the external field, which influences the saturation regime of the OTOC, a region crucial for characterizing the chaotic behavior of the system. To quantify the degree of chaoticity, we compute the normalized Fourier spectrum of the OTOC and observe that increasing the ramping field strength leads to a suppression of oscillation frequencies in the saturation region of the OTOC, thereby enhancing the system's chaotic behaviour. 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.