Splitting of the ground state manifold of classical Heisenberg spins as couplings are varied

We construct clusters of classical Heisenberg spins with two-spin $vec{S}_i.vec{S}_j$-type interactions for which the ground state manifold consists of disconnected pieces. We extend the construction to lattices and couplings for which the ground state manifold splits into an exponentially large number of disconnected pieces at a sharp point as the interaction strengths are varied with respect to each other. In one such lattice we construct, the number of disconnected pieces in the ground state manifold can be counted exactly.