  In this paper we discuss various correlations measured by the concurrence
(C), classical correlation (CC), quantum discord (QD), and geometric measure of
discord (GMD) in a two-qubit Heisenberg XXZ spin chain in the presence of
external magnetic field and Dzyaloshinskii-Moriya (DM) anisotropic
antisymmetric interaction. Based on the analytically derived expressions for
the correlations for the cases of thermal equilibrium and the inclusion of
intrinsic decoherence, we discuss and compare the effects of various system
parameters on the correlations in different cases. The results show that the
anisotropy Jz is considerably crucial for the correlations in thermal
equilibrium at zero temperature limit but ineffective under the consideration
of the intrinsic decoherence, and these quantities decrease as temperature T
rises on the whole. Besides, J turned out to be constructive, but B be
detrimental in the manipulation and control of various quantities both in
thermal equilibrium and under the intrinsic decoherence which can be avoided by
tuning other system parameters, while D is constructive in thermal equilibrium,
but destructive in the case of intrinsic decoherence in general. In addition,
for the initial state $|\Psi_1(0) > = \frac{1}{\sqrt{2}} (|01 > + |10 >)$, all
the correlations except the CC, exhibit a damping oscillation to a stable value
larger than zero following the time, while for the initial state $|\Psi_2(0) >
= \frac{1}{\sqrt{2}} (|00 > + |11 >)$, all the correlations monotonously
decrease, but CC still remains maximum. Moreover, there is not a definite
ordering of these quantities in thermal equilibrium, whereas there is a
descending order of the CC, C, GMD and QD under the intrinsic decoherence with
a nonnull B when the initial state is $|\Psi_2(0) >$.
