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Physical review. B, Condensed matter

Real-space renormalization-group investigation of the three-dimensional semi-infinite Blume-Emery-Griffiths model

A. Benyoussef, N. Boccara, M. El Bouziani

Published Dec 1, 1986• 33 citations
Publisher

Abstract

We investigate the three-dimensional semi-infinite Blume-Emery-Griffiths (BEG) model using the Migdal-Kadanoff renormalization-group method. The parameter space is six dimensional. We find 69 fixed points describing a large variety of critical behaviors and discuss various semi-infinite models which are special cases of the BEG model such as the Blume-Capel model. For this last model we determine the various generic types of phase diagrams according to the values of the interactions on the surface and in the bulk of the system.

DOI
10.1103/physrevb.34.7775

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We investigate the three-dimensional semi-infinite Blume-Emery-Griffiths (BEG) model using the Migdal-Kadanoff renormalization-group method. The parameter space is six dimensional. We find 69 fixed points describing a large variety of critical behaviors and discuss various semi-infinite models which are special cases of the BEG model such as the Blume-Capel model. For this last model we determine the various generic types of phase diagrams according to the values of the interactions on the surface and in the bulk of the system.
Using a renormalization-group transformation in position space based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite cubic spin-32\frac{3}{2}23​ Blume-Emery-Griffiths model with single-site uniaxial anisotropy and nearest-neighbor pair interactions, both bilinear and biquadratic. The fixed-point topologies and corresponding phase diagrams predicted for this model are presented. Attention is focused on the renormalization-group description of the surface critical behaviors, in comparison with the mean-field approximation results. Three main classes of system phase diagrams are determined, featuring a rich variety of phase transitions associated with the surface with type being according to the values of bulk and surface interaction parameters. Surface critical, tricritical, bicritical, and multicritical points are also found.
Phase transitions of the three-dimensional semi-infinite Blume-Emery-Griffiths model, extended to negative biquadratic interactions, are investigated within the framework of the mean-field approximation in comparison with the real-space renormalization-group results found previously. All the possible qualitative types of phase diagrams characterizing phase transitions of the model, including high entropy ferrimagnetic and staggered quadrupolar phases, have been determined. It is shown that such a system can exhibit critical and tricritical behavior on the surface and multicritical points.
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As a function of the ratio R of bulk and surface interactions and the ratio D of bulk and surface crystal field we have determined, within the framework both of the mean-field approximation and by renormalization-group techniques, various types of phase diagram of a three-dimensional semi-infinite ferromagnetic spin-3/2 Blume-Capel model. We found that there exist four main types of phase diagram showing a variety of phase transitions associated with the surface and unusual multicritical topologies, including ordinary, extraordinary, surface and special phase transitions. Moreover, in the ( R , D ) plane, we determined the domains in which the system exhibits a particular type of phase diagram.
By the use of the Migdal–Kadanoff renormalization group technique and the mean field approximation, we have explored the critical behavior of the semi-infinite mixed spin-7/2 and spin-1/2 Blume–Capel model. As a function of the computation ratios (bulk-surface) R and Y, different phase diagrams in the bulk and on the surface are classified and determined in the (surface anisotropy, temperature) plane. We have found four types of phase diagrams characterized by ordinary, extraordinary, surface, and special phase transitions. The derivative of the free energy and the behavior of the bulk and surface magnetizations are plotted at very low temperatures proving the existence of first-order transitions for both the surface and bulk. We have also presented the related fixed points and the critical exponents manifesting several classes of universality at the surface. Otherwise, a comparison was made between the two methods as well as with previous studies.
The Ising and Blume-Emery-Griffiths (BEG) models' critical behavior is analyzed in two dimensions and three dimensions by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under renormalization allow for the determination of the Néel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure does not yield a strong-disorder critical point corresponding to the transition to the spin-glass phase. This limitation is thoroughly analyzed and motivated.