Spin transport in an electrically driven magnon gas near Bose-Einstein condensation: Hartree-Fock-Keldysh theory

An easy-plane ferromagnetic insulator in a uniform external magnetic field and in contact with a phonon bath and a normal metal bath is studied theoretically in the presence of DC spin current injection via the spin Hall effect in the metal. The Keldysh path integral formalism is used to model the magnon gas driven into a nonequilibrium steady state by mismatched bath temperatures and/or electrical injection, and we analyze the magnon system in the normal state, but close to the instability to Bose-Einstein condensation, within the self-consistent Hartree-Fock approximation. We find that the steady state magnon distribution function generally has a nonthermal form that cannot be described by a single effective chemical potential and effective temperature. In the presence of two bath temperatures, we find that the correlation length of the superfluid order parameter fluctuations exhibits nontrivial finite temperature crossover behaviors that are richer than the standard crossover behaviors obtained for the vacuum-superfluid transition in an equilibrium dilute Bose gas. We study the consequences of these thermal crossovers on the magnon spin conductivity.