  The anisotropic spin-1/2 antiferromagnetic Heisenberg systems are studied on
three typical diamond-type hierarchical lattices (systems A, B and C) with
fractal dimensions df=1.63, 2 and 2.58, respectively. For system A, using the
real-space renormalization group approach, we calculate the phase diagram, the
critical exponent and quantum discord, and find that there exists a reentrant
behavior in the phase diagram. We also find that the quantum discord reaches
its maximum at T=0 and the thermal quantum discord decreases with the increase
of L, and it is almost zero at L>30. No matter how large the size of system is,
quantum discord will change to 0 when anisotropic parameter Delta=1. For
systems B and C, using the equivalent transformation and the real-space
renormalization group method, we obtain phase diagrams and find that: the Neel
temperature tends to zero in the isotropic Heisenberg limit on df=2 system;
there exists a phase transition in the isotropic Heisenberg model on system C.
By studying quantum discord, we find that there is a certain degree of twist of
the relation curve between quantum discord and T when Delta=0. Moreover, as an
example, we discuss the quantum effect in system A, which can be responsible
for the existence of the reentrant behavior in the phase diagram.
