Random field xy model in three dimensions

We study the random field xy spin model at T = 0 numerically on lattices of up to 1000 x 1000 x 1000 spins with emphasis on the weak random field. Our numerical method is physically equivalent to slow cooling in which the system is gradually losing energy and relaxing to an energy minimum. The system shows glass properties, the resulting spin states depending strongly on the initial conditions. Random initial condition for the spins leads to a vortex glass (VG) state with short-range spin-spin correlations defined by the average distance between vortex lines. Collinear and some other vortex-free initial conditions result in vortex-free ferromagnetic (F) states that have a lower energy. The energy difference between the F and VG states correlates with the vorticity of the VG state. The correlation functions in the F states agree with the Larkin-Imry-Ma theory at short distances. The hysteresis curves for a weak random field are dominated by topologically stable spin walls ruptured by vortex loops. We find no relaxation paths from the F, VG, or any other states to the hypothetical vortex-free state with zero magnetization.