Pairing, ferromagnetism, and condensation of a normal spin-1 Bose gas

We find the conditions under which the normal state of a spin-1 Bose gas is unstable toward condensation, ferromagnetism, liquid crystalline-like nematicity, and Bardeen-Cooper-Schrieffer-like pairing. When the spin-dependent interactions are much weaker than the density-density interaction there is direct transition from a featureless normal state to a fully ordered Bose-Einstein condensate with either ferromagnetic or nematic order. When the spin-independent and spin-dependent interactions are of comparable magnitude, we find several different symmetry breaking transitions at intermediate temperatures above the Bose-condensation transition temperature. We make predictions for the $T_c$ for these transitions, and assess the role of magnetic field and finite system size.