  We study ring-exchange models for bosons or XY-spins on the triangular
lattice. A four-spin exchange leads to a manifold of ground states with gapless
excitations and critical power-law correlations. With a nearest-neighbour
exchange, fluctuations select a four-fold ferrimagnetically ordered ground
state with a small spin/superfluid stiffness which breaks the global U(1) and
translational symmetry. We explore consequences for phase transitions at finite
temperature and in an in-plane magnetic field.
