Ferromagnetism, antiferromagnetism, and the curious nematic phase of S=1 quantum spin systems

We investigate the phase diagram of S=1 quantum spin systems with SU(2)-invariant interactions, at low temperatures and in three spatial dimensions. Symmetry breaking and the nature of pure states can be studied using random loop representations. The latter confirm the occurrence of ferro- and antiferromagnetic transitions and the breaking of SU(3) invariance. And they reveal the peculiar nature of the nematic pure states which MINIMIZE \sum_x (S_x^i)^2.