Berezinskii-Kosterlitz-Thouless transitions of an easy-plane ferromagnetic spin-1 Bose gas

In two-dimensional Bose gases thermal fluctuations preclude the formation of long-range order. Consequently, the superfluid transition is of the Berezinskii-Kosterlitz-Thouless (BKT) type, characterized by the emergence of quasi-long-range order driven by the binding of vortex-antivortex pairs. Spinor Bose gases provide a unique platform to study BKT transitions, on account of their ability to simultaneously exhibit phase and spin ordering. Here we utilize a stochastic Gross-Pitaevskii model to investigate the BKT transitions in an easy-plane ferromagnetic spin-1 Bose gas. We find that in general two BKT transitions occur with decreasing temperature: first mass superfluidity is established, followed at a lower temperature by spin superfluidity. We demonstrate that these transitions are associated with the binding of mass and spin vortex-antivortex pairs respectively. Furthermore, we develop a phase diagram dependent on quadratic Zeeman energy, which demonstrates that the mass and spin superfluid transitions become indistinguishable in the isotropic ferromagnetic phase.