Dynamics of magnetizations in the classical chaotic regime of the all-to-all Ising model with dissipation

We study the all-to-all Ising model in the presence of dissipation and periodic external field. The corresponding Lindblad equation has a time-periodic Liouvillian. The dynamics of magnetizations is explored by using both the mean-field theory and numerical simulation at a finite number of spins. The mean-field result exhibits a transition from the periodic response at small field amplitude to the chaotic dynamics at large amplitude. The Lyapunov exponents are calculated, which supports the existence of a classical chaotic phase. However, in the numerical simulation at a finite number of spins, the response is periodic at both small and large amplitudes. The scaling analysis of the Floquet Liouvillian spectrum suggests that the periodic response persists even in the thermodynamic limit. Further analysis shows that, in the classical chaotic regime, the density-matrix elements have a wide dispersion, instead of being localized. Such a delocalization of wave packets explains the failure of mean-field approximation.