Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Renormalisation group studies of the Blume-Emery-Griffiths model in two dimensions

Joan Adler, Amnon Aharony A, J. Oitmaa

Published May 1, 1978• 31 citations
Publisher

Abstract

The authors study a spin-1 Blume-Emery-Griffiths model on the triangular lattice using a generalisation of Niemeijer (1974) and van Leeuwen's (1975) discrete spin renormalisation group transformation which preserves the Potts symmetry of this model. An overall phase diagram is obtained that is basically in agreement with other treatments of this model. The exponents are compared with other renormalisation group studies and some series results.

DOI
10.1088/0305-4470/11/5/026

Related papers

Renormalisation group calculations for two dimensional disordered Ising models

Michael Plischke, D. Zobin

The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.

Journal of Physics C Solid State Physics•Nov 28, 1977•29 citations

Renormalisation group calculations on a mixed-spin system in two dimensions

S. L. Schofield, Roger Bowers

The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.

Journal of Physics A Mathematical and General•Dec 1, 1980•74 citations

Blume-Emery-Griffiths-Potts model in two dimensions: Phase diagram and critical properties from a position-space renormalization group

A. Nihat Berker, Michael Wortis

The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear ($J$) and biquadratic ($K$) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in $J$, $K$, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field ($H$) and another odd interaction ($L$) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.

Physical review. B, Solid state•Dec 1, 1976•371 citations

Renormalisation group approach for the two dimensional random Ising model

Gary S. Grest

The Niemeyer-van Leeuwen formulation of the renormalisation group is used to describe a quenched dilute Ising model on a triangular lattice in two dimensions. The two- and three-cluster approximations are applied to calculate the probability dependence of the critical temperature T c (p) and the critical exponents near the percolation threshold phi =1 is found to be in agreement with previous epsilon expansion results.

Journal of Physics C Solid State Physics•Apr 14, 1978•10 citations

Position-space renormalization-group investigation of the spin-3/2 Blume-Emery-Griffiths model with repulsive biquadratic coupling

A. Bakchich, M. El Bouziani

Phase transitions of a four-component lattice gas or spin-3/2 Blume-Emery-Griffiths model, with a single-ion uniaxial anisotropy and nearest-neighbour pair interactions, both bilinear and biquadratic, are investigated for two-dimensional lattices using an approximate renormalization-group approach of the Migdal-Kadanoff type. The set of fixed points and flows provide the characteristic phase diagrams, in the case of repulsive biquadratic interaction, featuring four ordered phases including high-entropy ferrimagnetic and staggered quadrupolar phases. Successive phase transitions and multicritical points are also found.

Journal of Physics Condensed Matter•Dec 6, 2000•25 citations

Small-cluster renormalization group in Ising and Blume-Emery-Griffiths models with ferromagnetic, antiferromagnetic, and quenched disordered magnetic interactions

Fabrizio Antenucci, A. Crisanti, Luca Leuzzi

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.

arXiv:1402.5820•Physical Review E•Jul 14, 2014•4 citations
The authors study a spin-1 Blume-Emery-Griffiths model on the triangular lattice using a generalisation of Niemeijer (1974) and van Leeuwen's (1975) discrete spin renormalisation group transformation which preserves the Potts symmetry of this model. An overall phase diagram is obtained that is basically in agreement with other treatments of this model. The exponents are compared with other renormalisation group studies and some series results.
The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.
The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
Phase transitions of a four-component lattice gas or spin-3/2 Blume-Emery-Griffiths model, with a single-ion uniaxial anisotropy and nearest-neighbour pair interactions, both bilinear and biquadratic, are investigated for two-dimensional lattices using an approximate renormalization-group approach of the Migdal-Kadanoff type. The set of fixed points and flows provide the characteristic phase diagrams, in the case of repulsive biquadratic interaction, featuring four ordered phases including high-entropy ferrimagnetic and staggered quadrupolar phases. Successive phase transitions and multicritical points are also found.
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.