Superfluid transition of a ferromagnetic Bose gas

The strongly ferromagnetic spin-1 Bose-Einstein condensate (BEC) has recently been realized with atomic $^{7}$Li. It was predicted that a strong ferromagnetic interaction can drive the normal gas into a magnetized phase at a temperature above the superfluid transition, and $^{7}$Li likely satisfies the criterion. We re-examine this theoretical proposal employing the two-particle-irreducible (2PI) effective potential, and conclude that there exists no stable normal magnetized phase for a dilute ferromagnetic Bose gas. For $^{7}$Li, we predict that the normal gas undergoes a joint first order transition and jump directly into a state with finite condensate density and magnetization. We estimate the size of the first order jump, and examine how a partial spin polarization in the initial sample affects the first order transition. We propose a qualitative phase diagram at fixed temperature for the trapped gas.