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In this paper, the phase diagrams and the critical behavior of the spin-1/2anisotropic XXZ ferromagnetic model (the anisotropic parameter{\Delta}\in(-\infty,1]) on two kinds of diamond-type hierarchical (DH) latticeswith fractal dimensions d_{f}=2.58 and 3, respectively, are studied via thereal-space renormalization group method. It is found that in the isotropicHeisenberg limit ({\Delta}=0), there exist finite temperature phase transitionsfor the two kinds of DH lattices above. The systems are also investigated inthe range of -\infty<{\Delta}<0 and it is found that they exhibit XY-like fixedpoints. Meanwhile, the critical exponents of the above two systems are alsocalculated. The results show that for the lattice with d_{f}=2.58, the value ofthe Ising critical exponent {\nu}_{I} is the same as that of classical Isingmodel and the isotropic Heisenberg critical exponent {\nu}_{H} is a finitevalue, 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 thequantum fluctuation at all temperatures and find the fluctuation of XY-likemodel is stronger than the anistropic Heisenberg model at the low-temperatureregion. By analyzing the fluctuation, we conclude that there will be remarkableeffect of neglecting terms on the final results of the XY-like model. However,we can obtain approximate result at bigger temperatures and give qualitativelycorrect picture of the phase diagram.

Phase diagrams and critical behavior of the quantum spin-1/2 XXZ model on diamond-type hierarchical lattices