Two-spin subsystem entanglement in spin 1/2 rings with long range interactions

We consider the two-spin subsystem entanglement for eigenstates of the Hamiltonian \[ H= \sum_{1\leq j< k \leq N} frac{1}{r_j,k}^α {\mathbf σ}_j\cdot {\mathbf σ}_k \] for a ring of N spins 1/2 with asssociated spin vector operator \(hbar /2\){bf σ}_j for the j-th spin. Here r_j,k is the chord-distance betwen sites j and k. The case α=2 corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for αneq 2. Two spin subsystem entanglement shows high sensistivity and distinguishes α=2 from αneq 2. There is no entanglement beyond nearest neighbors for all eigenstates when α=2. Whereas for αneq 2 one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of α which depends on the energy. The ground state (which is a singlet only for even N) does not have entanglement beyond nearest neighbors, and the nearest neighbor entanglement is virtually independent of the range of the interaction controlled by α.