  We analyze the phases of an S=1/2 spin model on a lattice of coupled
tetrahedra. The presence of both Heisenberg and antisymmetric,
Dzyaloshinsky-Moriya interactions can lead to two types of symmetry-broken
states: non-magnetic dimer order and, unexpectedly, exotic 4 sub-lattice weak
antiferromagnetic order - a state with a generically small ordered moment and
non-zero chirality. External magnetic field also induces weak
antiferromagnetism co-existing with strong dimer correlations in the ground
state.
  These states are formed as a result of broken Ising symmetries and exhibit a
number of unusual properties.
