  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 (uncondensed) state, but close to the instability to Bose-Einstein
condensation (BEC), within the self-consistent Hartree-Fock approximation. We
find that the steady state magnon distribution function generally has a
non-thermal form that cannot be described by a single effective chemical
potential and effective temperature. We also show that the BEC instability in
the electrically-driven magnon system is signaled by a sign change in the
imaginary part of the poles for long-wavelength magnon modes and by the
divergence of the nonequilibrium magnon distribution function. 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
and obtain an inverse square-root divergence in the spin conductivity in the
vicinity of the electrically-induced BEC instability. A spintronics device
capable of testing our spin transport predictions is discussed.
