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Pair entanglement in dimerized spin-s chains

We examine the pair entanglement in the ground state of finite dimerized spin-$s$ chains interacting through anisotropic $XY$ couplings immersed in a transverse magnetic field, by means of a self-consistent pair mean field approximation. The approach, which makes no a priori assumptions on the pair states, predicts, for sufficiently low coupling between pairs, $2s$ distinct dimerized phases for increasing fields below the pair factorizing field, separated by spin parity breaking phases. The dimerized phases lead to approximate magnetization and pair entanglement plateaus, while the parity breaking phases are characterized by weak pair entanglement but non-negligible entanglement of the pair with the rest of the system. These predictions are confirmed by the exact results obtained in finite $s=1$ and $s=3/2$ chains. It is also shown that for increasing values of the spin $s$, the entanglement of an isolated pair, as measured by the negativity, rapidly saturates in the anisotropic $XY$ case but increases as $s^{1/2}$ in the $XX$ case, reflecting a distinct single spin entanglement spectrum.