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 $\mathrm{\Phi }$ 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 $\mathrm{\Phi }$ being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for $\mathrm{\Phi }$ being a ground state. The classical case $s\to \mathrm{\infty }$ 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. $\mathrm{\Phi }$ 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.