Superfluid vs ferromagnetic behavior in a Bose gas of spin-1/2 atoms

We study the thermodynamic phases of a Bose gas of spin-1/2 atoms in the Hartree-Fock approximation. Our main result is that, for repulsive or weakly attractive inter-component interaction strength, the superfluid and ferromagnetic phase transitions occur at the same temperature. For strongly attractive inter-component interaction strength, however, the ferromagnetic phase transition occurs at a higher temperature than the superfluid phase transition. We also find that the presence of a condensate acts as an effective magnetic field that polarizes the normal cloud. We finally comment on the validity of the Hartree-Fock approximation in describing different phenomena in this system.