  The spin-1/2 antiferromagnetic Heisenberg systems are studied on three
typical diamond-type hierarchical lattices (systems A, B and C) with fractal
dimensions 1.63, 2 and 2.58, respectively, and the phase diagrams, critical
phenomena and quantum correlations are calculated by a combination of the
equivalent transformation and real-space renormalization group methods. We find
that there exist a reentrant behavior for system A and a finite temperature
transition in the isotropic Heisenberg limit for system C (not for system B).
Unlike the ferromagnetic case, the Neel temperatures of antiferromagnetic
systems A and B are inversely proportional to ln(Delta_c-Delta) (when
Delta->Delta_c) and ln Delta (when Delta->0), respectively. And we also find
that there is a turning point of quantum correlation in the isotropic
Heisenberg limit Delta=0 where there is a peak of the contour and no matter how
large the size of system is, quantum correlation will change to zero in the
Ising limit for the three systems. The quantum correlation decreases with the
increase of lattice size L and it is almost zero when L>=30 for system A, and
for systems B and C, they still exist when L is larger than that of system A.
Moreover, as an example, we discuss the error of result in system A, which is
induced by the noncommutativity.
