Stochastic Gross-Pitaevskii theory for a spin-1 Bose gas: Application to superfluidity in two dimensions

This paper develops and implements the stochastic projected Gross-Pitaevskii equation for spin-1 Bose gases, addressing key considerations for numerical simulations. As an application of the theory we explore equilibrium phases in a two-dimensional spin-1 gas, where quasi-long-range order emerges via a Berezinskii-Kosterlitz-Thouless transition. Our analysis includes definition of superfluid densities for both mass and spin degrees of freedom, in a manner suitable for implementation within a stochastic projected Gross-Pitaevskii equation simulation. We present a finite-temperature phase diagram for the ferromagnetic spin-1 Bose gas and identify three distinct superfluid phases: two exhibiting conventional Berezinskii-Kosterlitz-Thouless-like behavior and a novel phase that simultaneously supports independent mass and spin superflows. As temperature increases, the stability region of this novel phase shrinks. We provide characterization of the phase transitions through consideration of the spin-component densities and the unbinding of multiple types of vortices. This work provides a foundation for further studies of nonequilibrium and finite-temperature phenomena in spinor Bose gases.