Entanglement in Spin Chains and Lattices with Long-Range Ising-Type Interactions

We consider $N$ initially disentangled spins, embedded in a ring or $d$-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit $N\to \infty$. We provide a sufficient condition when bipartite entanglement between blocks of $L$ neighboring spins and the remaining system saturates and determine $S_L$ analytically for special configurations. We find an unbounded increase of $S_L$ as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large $N$, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.