A new microscopic representation of the spin dynamics in quantum systems with the Coulomb exchange interactions

There is a version of the Landau-Lifshitz equation that takes into account the Coulomb exchange interactions between atoms, expressed by the term $\sim\bm{s}\times\triangle\bm{s}$. On the other hand, ions in the magnetic materials have several valence electrons on the $d$-shell, and therefore the Hamiltonian of many-electron atoms with spins $S>1$ should include a biquadratic exchange interaction. We first propose a new fundamental microscopic derivation of the spin density evolution equation with an explicit form of biquadratic exchange interaction using the method of many-particle quantum hydrodynamics. The equation for the evolution of the spin density is obtained from the many-particle Schrodinger-Pauli equation and contains the contributions of the usual Coulomb exchange interaction and the biquadratic exchange. Furthermore, the derived biquadratic exchange torque in the spin density evolution equation is proportional to the nematic tensor for the medium of atoms with spin $\textit{S = 1}$. Our method may be very attractive for further studies of the magnetoelectric effect in multiferroics.