  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.
