We consider the two-spin subsystem entanglement for eigenstates of the Hamiltonian [ H= sum_{1leq j< k leq N} (frac{1}{r_{j,k}})^{alpha} {mathbf sigma}_jcdot {mathbf sigma}_k ] for a ring of $N$ spins 1/2 with asssociated spin vector operator $(hbar /2){bf sigma}_j$ 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 $alpha eq 2$. Two spin subsystem entanglement shows high sensistivity and distinguishes $alpha =2$ from $alpha eq 2$. There is no entanglement beyond nearest neighbors for all eigenstates when $alpha =2$. Whereas for $alpha eq 2$ 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 $alpha$.
Download