The Study Of Phase Transitions Of Quantum Spin Systems On Fractal Lattices

In this thesis, using the real-space renormalization group method, the phase transitions and critical behavior of the quantum spin model on diamond-type hierarchical (DH) lattices are studied. The main results of this thesis are as follows:1. The phase diagrams and the critical behavior of the spin-1/2 anisotropic XXZ ferromagnetic model (the anisotropic parameter Δ ∈ (-∞,1]) on DH lattices with fractal dimensions d_f = 2.58 and 3, respectively, and scaling factors b = 2 are studied. It is found that in the isotropic Heisenberg limit ( Δ = 0 ), there exist finite temperature phase transitions for the two kinds of DH lattices above. The systems are also investigated in the range of - ∞ < Δ < 0 and found that they exhibit XY-like fixed points. Meanwhile, the critical exponents of the above two systems are also calculated. For the lattice with d_f = 2.58, the value of the Ising critical exponent v_I is the same as that of classical Ising model on the same lattice, and the isotropic Heisenberg critical exponent v_H is a finite value (this is not the case for the two dimensional regular lattice, where v_H = ∞). For the lattice with d_f = 3, the values of v_I, and v_H agree well with those obtained on the simple cubic lattice.2. The phase transitions and critical behavior of the anisotropic Heisenberg system with Dzyaloshinsky-Moriya (DM) interactions on the DH lattice with d_f = 2 are studied. By numerically iterating the RG recurrent equations, the phase diagram, the critical points and critical exponents of the system are obtained. Compared with the known results for the Heisenberg system without DM interaction on the same lattice, the results indicate that the DM interaction does not change the critical behavior of the system. The two systems belong to asame universality class.