Quantum hydrodynamic representation of the exchange interaction in the theory of description of magnetically ordered media

Ferromagnets, multiferroics and other magnetically ordered materials are described by various models of the evolution of the magnetization of the medium. In this paper, we develop the method of many-particle quantum hydrodynamics for such media. We use the Heisenberg Hamiltonian and derive an equation for the evolution of macroscopic magnetization, corresponding to the non-dissipative version of the Landau-Lifshitz equation for particles with spin 1/2. It is shown that the well-known form of the contribution of the exchange interaction to the Landau-Lifshitz equation arises in the third order in terms of the interaction radius. The possibilities of the systematic generalization of the result obtained are discussed when considering the fifth order in the interaction radius or when considering particles with a large spin.