Dynamic phase transitions in the anisotropic $\mathrm{XY}$ spin system in an oscillating magnetic field

The Ginzburg-Landau model for the anisotropic $\mathrm{XY}$ spin system in an oscillating magnetic field below the critical temperature $T_c,$ $\dot{\psi }\left(\mathit{r}{,t)=(T}_c-T\right)\psi -|\psi {|}^2\psi +\gamma {\psi }^{*}+{\nabla }^2\psi +h\cos \left(\Omega t\right)$ is both theoretically and numerically studied. Here $\psi$ is the complex order parameter and $\gamma$ stands for the real anisotropy parameter. It is numerically shown that the spatially uniform system shows various characteristic oscillations (dynamical phases), depending on the amplitude h and the frequency $\Omega$ of the external field. As the control parameter, either h or $\Omega ,$ is changed, there exist dynamical phase transitions (DPT), separating them. By making use of the mode expansion analysis, we obtain the phase diagrams, which turn out to be in a qualitative agreement with the numerically obtained ones. By carrying out the Landau expansion, the reduced equations of motion near the DPT are derived. Furthermore, taking into account the spatial variation of order parameters, we will derive the analytic expressions for domain walls, which are represented by the Néel and Bloch type walls, depending on the difference of the coexistence of phases.