Characteristic, dynamic, and near-saturation regions of out-of-time-order correlation in Floquet Ising models

We study characteristic, dynamic, and saturation regimes of the out-of-time-order correlation (OTOC) in the constant-field Floquet system with and without longitudinal field. In the calculation of OTOC, we take local spins in longitudinal and transverse directions as observables which are local and nonlocal in terms of Jordan-Wigner fermions, respectively. We use the exact analytical solution of OTOC for the integrable model (without the longitudinal field term) with transverse direction spins as observables and numerical solutions for other integrable and nonintegrable cases. OTOCs in both cases depart from unity at a kick equal to the separation between the observables when the local spins are in the transverse direction, and one additional kick is required when the local spins are in the longitudinal direction. The number of kicks required to depart from unity depends on the separation between the observables and is independent of the Floquet period and system size. In the dynamic region, OTOCs show power-law growth in both models, the integrable one (without longitudinal field) and the nonintegrable one (with longitudinal field). The exponent of the power-law increases with increasing separation between the observables. Near the saturation region, OTOCs grow linearly at a very low rate.