Maximized string order parameters in the valence bond solid states of quantum integer spin chains

We propose a set of maximized string order parameters to describe the hidden topological order in the valence bond solid states of quantum integer spin-S chains. These optimized string order parameters involve spin-twist angles corresponding to ZS+1 rotations around z or x-axes, suggesting a hidden ZS+1 x ZS+1 symmetry. Our results also suggest that a local triplet excitation in the valence bond solid states carries a ZS+1 topological charge measured by these maximized string order parameters