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We compute the finite-temperature phase diagram of a pseudospin-$1/2$ Bosegas with contact interactions, using two complementary methods: the randomphase approximation (RPA) and self-consistent Hartree-Fock theory. We show thatthe inter-spin interactions, which break the (pseudo) spin-rotational symmetryof the Hamiltonian, generally lead to the appearance of a magnetically orderedphase at temperatures above the superfluid transition. In three dimensions, wepredict a normal easy-axis/easy-plane ferromagnet for sufficiently strongrepulsive/attractive inter-species interactions respectively. The normaleasy-axis ferromagnet is the bosonic analog of Stoner ferromagnetism known inelectronic systems. For the case of inter-spin attraction, we also discuss thepossibility of a \textit{bosonic} analog of the Cooper paired phase. This stateis shown to significantly lose in energy to the transverse ferromagnet in threedimensions, but is more energetically competitive in lower dimensions.Extending our calculations to a spin-orbit-coupled Bose gas with equal Rashbaand Dresselhaus-type couplings (as recently realized in experiment), weinvestigate the possibility of stripe ordering in the normal phase. Within ourapproximations however, we do not find an instability towards stripe formation,suggesting that the stripe order melts below the condensation temperature,which is consistent with the experimental observations of Ji \textit{et al.}[Ji \textit{et al.}, Nature Physics \textbf{10}, 314 (2014)].

Stoner ferromagnetism in a thermal pseudospin-1/2 Bose gas