  We clarify the condition for the occurrence of magnon Bose-Einstein
condensation (BEC) in spin pumping systems without using external pumping
magnetic fields. The Goldstone model is generalized and the stability of the
vacuum is closely investigated. By applying the generalized Goldstone model to
spin pumping systems, the condition for the experimental realization of the
stable magnon BEC in spin pumping systems is theoretically proposed.
