Superfluid vs Transverse Ferromagnetic Behaviour in a Bose Gas of Spin-1/2 Atoms

We study the thermodynamic phases of a 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. In recent years, studies of multi-component Bose-Einstein Condensates (BEC) have revealed a variety of interesting phenomena that reflect qualitatively different ideas from spinless condensates [1, 2, 3, 4, 5, 6, 7, 8]. Gases of spin-1/2 [1], spin-1 [2, 3, 4, 5, 6, 7] and spin-2 [7, 8] atoms have been studied in considerable detail, both experimentally and theoretically. New physical phenomena include fragmented states [4], coreless vortices [5] and complex spin dynamics [6, 7, 8]. Truely spin-1/2 particles are fermions. However, a gas of atoms each of which can occupy two different internal states can be treated as a gas of spin-1/2 atoms that obey Bose statistics (Note that in nonrelativistic systems, there is no a priori connection between spin and statistics). Examples of such gases are mixtures of Rb atomic gases in two hyperfine states and spin-polarized Hydrogen [1, 9]. It is well known that in a gas of noninteracting spin-1/2 bosonic atoms, the emergence of superfluid order is accompanied by the emergence of ferromagnetic order [10, 11]. A natural question that arises in that context is whether these two orders would remain connected if we include the effects of interatomic interactions. Perhaps a clearer way to pose the question is as follows: is it possible to have one of those two orders (i.e. superfluid or ferromagnetic) without having the other? One can give a number of elementary arguments that hint one way or the other. On the one hand, in the noninteracting system the connection between the two orders follows from the symmetry of the wave function and is not related to any thermodynamic arguments (except for saying that the normal cloud is unpolarized). That would suggest a rather robust connection between superfluid and ferromagnetic behaviours. On the other hand, if one considers the form of the order parameters or the symmetry of the Hamiltonian, there is no reason to believe that the two orders must be related. Keeping in mind certain caveats, the superfluid order parameter can be chosen as ∣