  In this paper, the phase diagrams and the critical behavior of the spin-1/2
anisotropic XXZ ferromagnetic model (the anisotropic parameter
{\Delta}\in(-\infty,1]) on two kinds of diamond-type hierarchical (DH) lattices
with fractal dimensions d_{f}=2.58 and 3, respectively, are studied via the
real-space renormalization group method. It is found that in the isotropic
Heisenberg limit ({\Delta}=0), there exist finite temperature phase transitions
for the two kinds of DH lattices above. The systems are also investigated in
the range of -\infty<{\Delta}<0 and it is found that they exhibit XY-like fixed
points. Meanwhile, the critical exponents of the above two systems are also
calculated. The results show that for the lattice with d_{f}=2.58, the value of
the Ising critical exponent {\nu}_{I} is the same as that of classical Ising
model and the isotropic Heisenberg critical exponent {\nu}_{H} is a finite
value, and for the lattice with d_{f}=3, the values of {\nu}_{I} and {\nu}_{H}
agree well with those obtained on the simple cubic lattice. We also discuss the
quantum fluctuation at all temperatures and find the fluctuation of XY-like
model is stronger than the anistropic Heisenberg model at the low-temperature
region. By analyzing the fluctuation, we conclude that there will be remarkable
effect of neglecting terms on the final results of the XY-like model. However,
we can obtain approximate result at bigger temperatures and give qualitatively
correct picture of the phase diagram.
