Diverging Length Scale, Scaling, and Universality of Entanglement Near a Quantum Phase Transition

In this work, we show that an important quantity to study about entanglement near a quantum phase transition is the two-body entanglement S(i,j), which measures the entanglement between two separated degrees of freedom (ij) and the rest of system. We establish its relation to correlation functions in the long range limit. Away from the critical point, S(n) saturates with a characteristic length scale $\xi_E$, as the distance n increases. The entanglement length $\xi_E$ diverges near the critical point with the same critical exponent as correlation length. At the critical point, S(n) follows a power law. The universality and finite size scaling of entanglement are demonstrated in a class of exactly solvable spin model.