  It is investigated theoretically that magnetized Bose-Einstein condensation
(BEC) with the internal (spin) degrees of freedom exhibits a rich variety of
phase transitions, depending on the sign of the interaction in the spin
channel. In the antiferromagnetic interaction case there exist always double
BEC transitions from single component BEC to multiple component BEC. In the
ferromagnetic case BEC becomes always unstable at a lower temperature, leading
to a phase separation. The detailed phase diagram for the temperature vs the
polarization, the spatial spin structure, the distribution of non-condensates
and the excitation spectrum are examined for the harmonically trapped systems.
