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 $$H=\sum _{1\le j<k\le N}(\frac{1}{r_{j,k}}{)}^{\alpha }{\mathbf{\sigma }}_j\cdot {\mathbf{\sigma }}_k$$ for a ring of $N$ spins 1/2 with asssociated spin vector operator $(\mathrm{\hslash }\mathrm{/}2){\mathbf{\sigma }}_j$ for the $j$-th spin. Here $r_{j,k}$ is the chord-distance betwen sites $j$ and $k$. The case $\alpha =2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\alpha \mathrm{\ne }2$. Two spin subsystem entanglement shows high sensistivity and distinguishes $\alpha =2$ from $\alpha \neq 2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\alpha =2$. Whereas for $\alpha \mathrm{\ne }2$ one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of $\alpha$ 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 $\alpha$.