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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices

Le Jian-Xin, Zhao‐Di Yang

Published Dec 17, 2003• 8 citations
Publisher

Abstract

The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.

DOI
10.1103/physreve.68.066105

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The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.
Using the real-space renormalization-group transformation, we study the phase transitions of the Ashkin-Teller model including the antiferromagnetic interactions on a type of diamond hierarchical lattices, of which the number of bonds per branch of the generator is odd. The isotropic Ashkin-Teller model and the anisotropic one are, respectively, investigated. We find that the phase diagram, for the isotropic Ashkin-Teller model, consists of five phases, two of which are associated with the partially antiferromagnetic ordering of the system, while the phase diagram, for the anisotropic Ashkin-Teller model, contains 11 phases, six of which are related to the partially antiferromagnetic ordering of the system. The correlation length critical exponents and the crossover exponents are also calculated.
Using the recursion method, we study the phase transitions of the Ashkin–Teller model on the Bethe lattice, restricting ourselves to the case of ferromagnetic interactions. The isotropic Ashkin–Teller model and the anisotropic one are respectively investigated, and exact expressions for the free energy and the magnetization are obtained. It can be found that each of the three varieties of phase diagrams, for the anisotropic Ashkin–Teller model, consists of four phases, i.e., the fully disordered paramagnetic phase Para , the fully ordered ferromagnetic phase Ferro , and two partially ordered ferromagnetic phases ⟨σ⟩ and ⟨σ s ⟩, while the phase diagram, for the isotropic Ashkin–Teller model, contains three phases, i.e., the fully disordered paramagnetic phase Para , the fully ordered ferromagnetic phase Baxter Phase , and the partially ordered ferromagnetic phase ⟨σ s ⟩.
In this paper, the phase diagrams and the critical behavior of the spin-1/2 anisotropic XXZ ferromagnetic model (the anisotropic parameter Δ\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 (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 -\infty
With a real-space renormalization-group method, the anisotropic quantum Heisenberg model on three-dimensional diamond-type hierarchical lattice is studied, and the phase diagram and critical properties are obtained. For the ferromagnetic system, it is shown that there is a finite-temperature phase transition for Δ=0 where Δ is the anisotropy parameter. The order parameter and critical exponents are also calculated. For the antiferromagnetic model, we find that the critical temperature is not equal to zero for Δ=0 and there is not reentrant behavior on the critical line.
By using Monte Carlo simulation,we have studied phase transitions of the isotropic Ashkin-Teller model,for the ferromagnetic case,on a two-dimensional rewiring-type small world network,and obtained phase diagrams for several different values of the rewiring probability.It can be found that the rewiring probability have little effect on the structure of the phase diagrams,each of which consists of three phases,i.e.,the fully disordered paramagnetic phase, the partially ordered ferromagnetic phase and the fully ordered ferromagnetic phase,and there are first-order transitions between the fully ordered ferromagnetic phase and the fully disordered paramagnetic phase as well as between the fully ordered ferromagnetic phase and the partially ordered ferromagnetic phase.
We have investigated the three-color Ashkin-Teller model (3AT), on the Wheatstone bridge hierarchical lattice, by means of a Migdal-Kadanoff renormalization group approach. We have obtained the exact recursion relations for the renormalized couplings, which have been used to investigate the phase diagram and to study the corresponding critical points. The phase diagram, represented in terms of the dual transmissivity vector, presents four magnetic phases and nine critical points. We have also numerically calculated the correlation length (νTν_TνT​) and crossover (ϕϕϕ) critical exponents, which show that seven of the critical points are in the Potts model universality class (q=2q=2q=2, 4 e 8). The remaining critical points are in a universality class which may belong to the Baxter's line. Our results are exact on the hierarchical lattice used in the present work and the phase diagram can be considered as an approximation to more realistic Bravais lattices.