Dynamics of a vector spin-glass model

We examine a model of spin-½ vectors randomly distributed and interacting via a $\pm r^{-3\mathrm{}}$ law with random signs. Here $r$ is the interspin distance. A mean-field theory is derived from the variational principle. Strongly coupled pairs, trios, and aggregates of more spins appear as more complicated single spins. The concept of local fields, including anisotropy fields emerges naturally. Statistics of local fields and metastable states are estimated. The magnetic susceptibility and the remanent magnetization at low temperatures are studied. Their time dependence and dependence on the applied field are calculated approximately.