The Phase Transtions Of Quantum Heisenberg Model On The Diamond-type Lattices

The anisotropic Heisenberg model is used to study the phase transitions and describe the magnetic material. In this paper, using the real-space renormalization group method, we study the ferromagnetic and antiferromagnetic Heisenberg model on a type of diamond-type hierarchical lattices.We study the phase transition and criticality of ferromagnetic anisotropic Heisenberg model on the three-dimensional diamond-type hierarchical lattice. The phase diagram of the system is obtained. It is shown that this system has a finite critical temperature forΔ= 0.We find that forΔ= 0 the system belongs to the universality class of Heisenberg, for 0 <Δ< 1 the universality class of this system is same as Ising model. Using the renormalization group method in the region far from the critical points, the order parameter of this system is studied. We find that it is obviously different from the two-dimensional hierarchical lattice. ForΔ= 0, the magnetization isn't equal to zero. This is in agreement with the results of phase diagram. We calculate the correlation length critical and magnetic critical exponents forΔ=1, v =1.06482,β=0.463; forΔ= 0, v = 1.51091,β= 1.Using the equivalent transformation method and the real-space renormalization method, we obtain the phase diagram of the antiferromagnetic Heisenberg model on the three-dimensional diamond-type hierarchical lattice. Comparing the results with ferromagnetic system, we find that the antiferromagnetic critical line is up on the ferromagnetic critical line, and the critical temperature isn't equal to zero. There isn't a reentrant on the critical line. It is different from the results of the model on the two-dimensional hierarchical lattice but in agreement with the results of the mean-field renormalization method.Using Migdal-Kadanoff renormalization method, we study anisotropic Heisenberg model on a type of diamond-type hierarchical lattices and the phase diagrams are obtained. The lower dimension of this system is obtained d_?= 2.26.The antiferromagnetic anisotropic Heisenberg model on a type of diamond-type hierarchical lattices is also studied and the phase diagrams are obtained. Comparing with phase diagrams of the ferromagnetic system, we find that the phase diagram of the diamond-type hierarchical lattice d_f= 2.26 is different from the results of the system on d_f= 2,2.58 and 3 hierarchical lattices. The ferromagnetic and antiferromagnetic critical lines are crossover. It is shown that fractal dimension of diamond-type hierarchical lattices make more effective to the antiferromagnetic system than the ferromagnetic system.