  We numerically investigate classical and quantum correlations in
one-dimensional quantum critical systems. The infinite matrix product state
(iMPS) representation is employed in order to consider an infinite-size spin
chain. By using the infinite time-evolving block decimation algorithm, iMPS
ground state wave functions are obtained at critical points for the
transverse-field spin-$1/2$ XY model. From the ground state wave functions, we
calculate classical and quantum correlations and mutual information. All of the
correlations are found to exhibit a power-law decay with the increments of the
lattice distance for both the transition lines of the Ising universality class
and the Gaussian universality class. Such power-law scaling behaviors of the
correlations manifest the existence of diversing correlation lengths, which
means scale invariance. The critical features of the correlations can be
characterized by introducing a critical exponent of the power-law decaying
correlations. Similar to the critical exponent $\eta$ of the spin-spin
correlation for the universality classes in the transverse-field XY model, we
calculate the critical exponents of the two-spin classical and quantum
correlations as well as that of the corresponding mutual information. All of
the correlations have the same critical exponents, i.e.,
$\eta^{I}=\eta^{C}=\eta^{D}$ at a critical point, where the superscripts $I$,
$C$, and $D$ stand for mutual information, classical correlation, and quantum
correlation, respectively. Furthermore, the critical exponent $\eta$ of the
spin-spin correlation is shown to relate to $\eta = \eta^\alpha/2$ with $\alpha
\in \{ I, C, D\}$.
