  We discuss the behavior of quantum and classical pairwise correlations in
critical systems, with the quantumness of the correlations measured by the
quantum discord. We analytically derive these correlations for general real
density matrices displaying $Z_2$ symmetry. As an illustration, we analyze both
the XXZ and the transverse field Ising models. Finite-size as well as infinite
chains are investigated and the quantum criticality is discussed. Moreover, we
identify the spin functions that govern the correlations. As a further example,
we also consider correlations in the Hartree-Fock ground state of the
Lipkin-Meshkov-Glick model. It is then shown that both classical correlation
and quantum discord exhibit signatures of the quantum phase transitions.
