Thermal entanglement of spins in mean-field clusters

We determine thermal entanglement in mean field clusters of $N$ spin one-half particles interacting via the anisotropic Heisenberg interaction, with and without external magnetic field. For the $xxx$ cluster in the absence of magnetic field we prove that only the N=2 ferromagnetic cluster shows entanglement. An external magnetic field $B$ can only entangle $xxx$ anti-ferromagnetic clusters in certain regions of the $B-T$ plane. On the other hand, the $xxz$ clusters of size $N>2$ are entangled only when the interaction is ferromagnetic. Detailed dependence of the entanglement on various parameters is investigated in each case.