Polarization evolution equation for exchange-strictionally formed type II multiferroic materials

 Multiferroics are materials where a single cell of a magnetically ordered crystal forms an electric dipole moment. In this work, we derive an equation for the evolution of the macroscopic density of the electric dipole moment (polarization of the system). For this purpose, we employ the quantum hydrodynamic method, which allows us to derive equations for the evolution of the macroscopic functions of quantum systems starting with the microscopic description. Here, we do not consider the microscopic level of individual electrons and ions; rather, we start our analysis from the combination of ions in the cells of the crystal. We present an effective Hamiltonian for the evolution of such intermediate-scale objects. We also apply the equation for the electric dipole moment of the single cell, which is re-contracted in the corresponding operator. Using this operator and the wave function of the combination of ions in the cell, we define the macroscopic density of the electric dipole moment. Finally, we apply the nonstationary Schrodinger equation with the chosen Hamiltonian in order to derive the equation for the evolution of the polarization which describes the dipole formed by the exchange-striction mechanism in type II multiferroic materials. The interaction-defined term in the polarization evolution equation is found to be proportional to the fourth space derivative of a mixed product (triple scalar product) of three spin-density vectors. Conditions are discussed for the regime where the interaction appears in a smaller order on the space derivatives.