Gauge Fields in the Theory of Condensed Matter and Helicity Conservation
A model of the quantum hydrodynamics of a charged particle in an external electromagnetic field is considered with account for the spin-orbit interaction and the influence of the Zeeman energy. The model is based on the formalism of introducing non-Abelian Yang-Mills gauge fields. It is shown that non-Abelian gauge fields generate forces, that act on the spin and spin current. It is demonstrated that the helicity of a particle moving on the background of such fields, that create forces that depend on the spin of the particle, is not conserved. A two- fluid model of quantum hydrodynamics is developed for systems of electrons in spin-up and spin-down states moving against the background of the magnetic structure of a skyrmion with a nontrivial topology. A system of continuity equations, momentum balance, and evolution equations of spin density is constructed, and the helicity conservation for electron fluxes is proved in the framework of the proposed two-fluid model.