Quantum critical point and entanglement in a matrix-product ground state

In this paper, we study the entanglement properties of a spin-1 model, the exact ground state of which is given by a matrix product (MP) state. The model exhibits a critical point transition at a parameter value a=0. The longitudinal and transverse correlation lengths are known to diverge as a?0. We use three different entanglement measures S(i) (the one-site von Neumann entropy), S(i,j) (the two-body entanglement), and G(2,n) (the generalized global entanglement) to determine the entanglement content of the MP ground state as the parameter a is varied. The entanglement length, associated with S(i,j), is found to diverge in the vicinity of the quantum critical point a=0. The first derivative of the entanglement measure E [=S(i),S(i,j)] with respect to the parameter a also diverges. The first derivative of G(2,n) with respect to a does not diverge as a→0 but attains a maximum value at a=0. At the quantum critical point itself all three entanglement measures become zero. We further show that multipartite correlations are involved in the quantum phase transitions at a=0