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Journal of Anqing Teachers College

Renormalization Group of the Gauss Model on a Type of Hierarchical Lattice

Zhu Zu-song

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Gauss mode is allowed to take a continuous spin value,so the study of Gauss mode is of great value to understand better the critical properties of ferromagnets.In this paper,using the renormalization-group transformation technique,the phase transition and critical properties of the modified Gauss model on a type of hierarchical lattice are studied,and its fixed points is K=b33 obtained.In terms of the RG transformation and scaling theory,critical exponents α=0.758 and v=0.414 are obtained.

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Gauss mode is allowed to take a continuous spin value,so the study of Gauss mode is of great value to understand better the critical properties of ferromagnets.In this paper,using the renormalization-group transformation technique,the phase transition and critical properties of the modified Gauss model on a type of hierarchical lattice are studied,and its fixed points is K=b33 obtained.In terms of the RG transformation and scaling theory,critical exponents α=0.758 and v=0.414 are obtained.
Using the renormalization-group transformation method together with the cumulative expansion technique,the phase transition and critical properties of the S4 model on a type of hierarchical lattice were studied,and its fixed points and critical exponents were obtained.The results show that there exists a Wilson-Fisher fixed point besides the Gaussian fixed point,compared with those of the S4 model on the special diamond-type of hierarchical lattices,the critical points and critical exponents have changed.It has been found that both of them belong to different university classes.
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.
Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter bbb. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of bbb around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.
The renormalization group (RG) methods are still far from being completely understood in quenched disordered systems. In order to gain insight into the nature of the phase transition of these systems, it is common to investigate simple models. In this work we study a real-space RG transformation on the Dyson hierarchical lattice with a random field, which leads to a reconstruction of the RG flow and to an evaluation of the critical exponents of the model at T=0. We show that this method gives very accurate estimations of the critical exponents by comparing our results with those obtained by some of us using an independent method.
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.