Thermal entanglement in fully connected spin systems and its RPA description

We examine the thermal pairwise entanglement in a symmetric system of $n$ spins fully connected through anisotropic $XYZ$-type couplings embedded in a transverse magnetic field. We consider both the exact evaluation together with that obtained with the static path + random phase approximation (RPA) and the ensuing mean field + RPA. The latter is shown to provide an accurate analytic description of both the parallel and antiparallel thermal concurrence in large systems. We also analyze the limit temperature for pairwise entanglement, which is shown to increase for large fields and to decrease logarithmically with increasing $n$. Special finite size effects are as well discussed.