Nonequilibrium dynamical phase transition of 3D kinetic Ising/Heisenberg spin system

We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. $T_{\rm TCP}=1.33\times\exp(-\omega/30.7)$. In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time $\tau$ depends upon the amplitude $H_0$, reduced temperature $T/T_{\rm C}$ and the frequency $\omega$ as $\tau=C\cdot\omega^{\alpha}\cdot H_0^{-\beta}\cdot(T/T_{\rm C})^{-\gamma}$.