  We study the classical correlation (CC) and quantum discord (QD) between two
spin subgroups of the Lipkin-Meshkov-Glick (LMG) model in both binary and
trinary decompositions of spins. In the case of bipartition, we find that the
classical correlations and all the quantum correlations including the QD, the
entanglement of formation (EoF) and the logarithmic negativity (LN) are
divergent in the same singular behavior at the critical point of the LMG model.
In the case of tripartition, however, the classical correlation is still
divergent but all the quantum correlation measures remain finite at the
critical point. The present result shows that the classical correlation is very
robust but the quantum correlation is much frangible to the environment
disturbance. The present result may also lead to the conjecture that the
classical correlation is responsible for the singularity behavior of physics
quantities at critical points of a many-body quantum system.
