Dynamical models and the phase ordering kinetics of the s=1 spinor condensate

The $s=1$ spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We investigate the formation of magnetic domains at finite temperature and magnetic field in two dimensions in an optical trap. We study the general ground state phase diagram of a spin-1 system and focus on a phase that has a magnetic Ising order parameter and numerically determine the nature of the finite temperature superfluid and magnetic phase transitions. We then study three different dynamical models: model A, which has no conserved quantities, model F, which has a conserved second sound mode and the Gross-Pitaevskii (GP) equation which has a conserved density and magnetization. We find the dynamic critical exponent to be the same for models A and F ($z=2$) but different for GP ($z approx 3$). Externally imposed magnetization conservation in models A and F yields the value $z approx 3$, which demonstrates that the only conserved density relevant to domain formation is the magnetization density.