Monogamy of entanglement and improved mean-field ansatz for spin lattices

We consider rather general spin-$1/2$ lattices with large coordination numbers $Z$. Based on the monogamy of entanglement and other properties of the concurrence $C$, we derive rigorous bounds for the entanglement between neighboring spins, such as $C\leq 1/\sqrt{Z}$, which show that $C$ decreases for large $Z$. In addition, the concurrence $C$ measures the deviation from mean-field behavior and can only vanish if the mean-field ansatz yields an exact ground state of the Hamiltonian. Motivated by these findings, we propose an improved mean-field ansatz by adding entanglement.