  Multipartite entanglement, measured by the geometric entanglement(GE), is
discussed for integer spin Valance-Bond-Solid (VBS) state respectively with
periodic boundary condition(PBC) and open boundary condition(OBC) in this
paper. The optimization in the definition of geometric entanglement can be
reduced greatly by exploring the symmetry of VBS state, and then the fully
separable state can be determined explicitly. Numerical evaluation for GE by
the random simulation is also implemented in order to demonstrate the validity
of the reductions. Our calculations show that GE is saturated by a finite value
with the increment of particle number, that means that the total entanglement
for VBS state would be divergent under the thermodynamic limit. Moreover it is
found that the scaling behavior of GE with spin number $s$ is fitted as
$\alpha\log (s+\tfrac{\beta}{s}+\gamma)+\delta$, in which the values of the
parameters $\alpha, \beta, \gamma, \sigma$ are only dependent on the parity of
spin $s$. A comparison with entanglement entropy of VBS state is also made, in
order to demonstrate the essential differences between multipartite and
bipartite entanglement in this model.

Quantum Physics