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 infinity. 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.