Electric polarization evolution equation for antiferromagnetic multiferroics with the polarization proportional to the scalar product of the spins

The spin current model of electric polarization of multiferroics is justified via the quantum hydrodynamic method and the mean-field part of the spin-orbit interaction. The spin current model is applied to derive the electric polarization proportional to the scalar product of the spins of the near by ions, which appears to be caused by the Dzylaoshinskii-Moriya interaction. Symmetric tensor spin structure of the polarization is discussed as well. We start our derivations for the ferromagnetic multiferroic materials and present the further generalization for the antiferromagnetic multiferroic materials. We rederive the operator of the electric dipole moment, which provides the macroscopic polarization obtained via the spin current model. Finally, we use the quantum average of the found electric dipole moment operator in order to derive the polarization evolution equation for the antiferromagnetic multiferroic materials. Possibility of spiral spin structures is analyzed.