Study Of Phase Transitions In Quantum Antiferromagnetic Heisenberg Systems On Fractal Lattices

Phase transitions and the critical behavior of quantum antiferromagnetic(AF)spin systems are very important study fields in condensed matter physics.With the development of this study,Wilson developed the renormalization group method in the 1970s.In recent years,the quantum spin systems on fractals increases interest in this model.In this thesis, using the renormalization group method,phase transitions and critical behavior of the Heisenberg AF model on three kinds of diamond-type hierarchical(DH)lattices are studied. The main results are as follows:1.The phase diagram and critical behavior of the spin-1/2 anisotropic AF Heisenberg model(the anisotropic parameterΔ∈[0,1])on the DH lattice with dimension d_f=1.63 are studied.The results indicate that there exist a critical valueΔ_c=0.703,at which the critical temperature goes to zero,and a reentrant behavior on the critical line between the ordered and disordered phases.For the Ising limit(Δ=1),the reduced critical temperature in the ferromagnetic(F)system is the same as that in the AF system.Moreover,it is found that the critical temperature for the F system behaves as T_c～ΔwhenΔ→0,and the Néel temperature T_N behaves as T_N～1/ln(Δ_c-Δ)whenΔis nearΔ_c.2.Using the equivalent transformation and the real-space renormalization group method,the phase diagrams of the anisotropic AF Heisenberg systems on DH lattices with d_f=2 and d_f= 2.58 are studied.The results show that:for the isotropic Heisenberg limit(Δ=0),the critical temperature tends to zero for the lattice with d_f=2,however, there exists phase transition on the d_f=2.58 lattice.Moreover,the reduced critical temperature in the F system is the same as that in the AF system for both lattices. Furthermore,for the d_f= 2 system,we also calculate the critical behavior when T→0. It is found that T_N behavesas T_N～1/ln|Δ_c-Δ|(Δ_c=0).3.The effects of quantum fluctuation in the phase transitions are discussed.The errors of the critical temperature and the anisotropy for F and AF systems are calculated.The results show that:in the T→∞limit,all errors tencl to zero;with the decrease of the anisotropy parameter,the effect of the fluctuation will be strengthened;at low temperatures, the fluctuation of the AF system is stronger than that of the F one.