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 $a=0$.