Superfluid transition of a ferromagnetic Bose gas

The strongly ferromagnetic spin-1 Bose-Einstein condensate has recently been realized with atomic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mmultiscripts><a:mi>Li</a:mi><a:mprescripts/><a:none/><a:mn>7</a:mn></a:mmultiscripts></a:math>. 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mmultiscripts><b:mi>Li</b:mi><b:mprescripts/><b:none/><b:mn>7</b:mn></b:mmultiscripts></b:math> likely satisfies the criterion. We reexamine this theoretical proposal employing the two-particle-irreducible effective potential, and conclude that there exists no stable normal magnetized phase for a dilute ferromagnetic Bose gas. For <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mmultiscripts><c:mi>Li</c:mi><c:mprescripts/><c:none/><c:mn>7</c:mn></c:mmultiscripts></c:math>, we predict that the normal gas undergoes a joint first-order transition and jumps 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.