In how many ways can four vectors add to zero?

Synopsis

A fundamental motif in frustrated magnetism is the tetrahedral cluster -- four spins with each pair separated by the same distance. If the spins are coupled by Heisenberg couplings, they must add to zero to minimize energy. This leads to a simple mathematical criterion for classical ground states -- we have four vectors which are constrained to add to zero. We show that this leads to a five-dimensional space of allowed states. Remarkably, this space has 'non-manifold' structure. It contains 'singular' points about which it appears to be six dimensional. We use this construction to build a semi-classical theory for the tetrahedral cluster. In the low-energy limit, this cluster takes a very simple form. It decomposes into two independent objects -- a rigid rotor (a spinning top) and a free spin. The free spin is perhaps the simplest example of an 'emergent' quantity, arising from angular variables in the spin configuration.  This provides an elegant way to understand the energy spectrum and physical properties of tetrahedral molecular magnets.