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=∑_1≤j<k≤N(1/r_j,k)^ασ_j⋅σ_k for a ring of N spin-1/2 particles with associated spin vector operator (ℏ/2)σ_j for the jth spin. Here r_j,k is the chord distance between sites j and k. The case α=2 corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for α≠2. Two-spin-subsystem entanglement shows high sensitivity and distinguishes α=2 from α≠2. There is no entanglement beyond nearest neighbors for all eigenstates when α=2. Whereas for α≠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.