Exact ground states for coupled spin trimers

We consider a class of geometrically frustrated Heisenberg spin systems which admit exact ground states. The systems consist of suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers s and their ground state Phi will be the product state of the local singlet ground states of the trimers. We provide linear equations for the inter-trimer coupling constants which are equivalent to Phi being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for Phi being a ground state. The classical case s -> infinity can be completely analyzed. For the quantum case we consider a couple of examples, where the critical values of the inter- trimer couplings are numerically determined. These examples include chains of corner sharing tetrahedra as well as certain spin tubes. Phi is proven to be gapped in the case of trimer chains. This follows from a more general theorem on quantum chains with product ground states.