Entanglement of finite cyclic chains at factorizing fields

We examine the entanglement of cyclic spin-1/2 chains with anisotropic XYZ Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually nonzero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization jump. Illustrative exact results are shown for chains with (1) full range and (11) nearest-neighbor couplings. Global entanglement properties tit such points are also discussed.