Weak antiferromagnetism and dimer order in quantum systems of coupled tetrahedra

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.