We consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian $H={\sum }_{1\le j<k\le N}{\left(\frac{1}{r_{j,k}}\right)}^{\alpha }{\sigma }_j\cdot {\sigma }_k$ for a ring of $N$ spin-1/2 particles with associated spin vector operator $\left(\hslash /2\right){\sigma }_j$ for the $j$th spin. Here $r_{j,k}$ is the chord distance between sites $j$ and $k$. The case $\alpha =2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\alpha \ne 2$. Two-spin-subsystem entanglement shows high sensitivity and distinguishes $\alpha =2$ from $\alpha \ne 2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\alpha =2$. Whereas for $\alpha \ne 2$ 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 $\alpha$.

• Received 13 September 2007

DOI:https://doi.org/10.1103/PhysRevA.77.022109