  For the identification of non-trivial quantum phase, we exploit a Bell-type
correlation that is applied to the one-dimensional spin-1 XXZ chain. It is
found that our generalization of bipartite Bell correlation can take a
decomposed form of transverse spin correlation together with high-order terms.
The formulation of density-matrix renormalisation group is utilized to obtain
the ground state of a given Hamiltonian with non-trivial phase. Subsequently
Bell-SLK-type generalized correlation is evaluated through the analysis of the
matrix product state. Diverse classes of quantum phase transitions in the
spin-1 model are identified precisely through the evaluation of the first and
the second moments of the generalized Bell correlations. The role of high-order
terms in the criticality has been identified and their physical implications
for the quantum phase has been revealed.
