The phase diagram of 2D polar condensates in a magnetic field

Spin one condensates in the polar (antiferromagnetic) phase in two dimensions are shown to undergo a transition of the Ising type, in addition to the expected Kosterlitz--Thouless (KT) transition of half vortices, due to the quadratic Zeeman effect. We establish the phase diagram in terms of temperature and the strength of the Zeeman effect using Monte Carlo simulations. When the Zeeman effect is sufficiently strong the Ising and KT transitions merge. For very strong Zeeman field the remaining transition is of the familiar integer KT type.