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

Abstract

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 $α\neq 2$. Two spin subsystem entanglement shows high sensistivity and distinguishes $α=2$ from $α\neq 2$. There is no entanglement beyond nearest neighbors for all eigenstates when $α=2$. Whereas for $α\neq 2$ 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 $α$.