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

Physical review. B, Condensed matter

Phase diagram for the antiferromagnetic Blume-Capel model near tricriticality

J. D. Kimel, Per Arne Rikvold, Yung‐Li Wang

Published Apr 1, 1992• 56 citations
Publisher

Abstract

We study the two-dimensional, antiferromagnetic Blume-Capel model on a square lattice by numerical transfer-matrix and Monte Carlo finite-size scaling methods, in order to investigate whether the line of tricritical points in this model may be decomposed into lines of critical end points and double critical points, as predicted by mean-field theory and recent Monte Carlo simulations on a three-dimensional cubic lattice. Conclusive numerical evidence is obtained, indicating that such decomposition does u/Inot occur in this two-dimensional model. The nondecomposition is explained in terms of the large fluctuations in the two-dimensional nearest-neighbor model, and we speculate that the situation may be different in two-dimensional antiferromagnetic models with weak, ferromagnetic next-nearest-neighbor interactions.

DOI
10.1103/physrevb.45.7237

Related papers

Monte Carlo renormalization-group study of tricritical behavior in two dimensions

David Paul Landau, Robert H. Swendsen

Using a Monte Carlo renormalization-group (MCRG) method we study tricritical behavior in two very different two-dimensional spin models: the Blume-Capel ferromagnet and the next-nearest-neighbor Ising antiferromagnet. The MCRG method is used to locate accurately tricritical points independent of the convergence of eigenvalue estimates. Despite the different symmetries and the different renormalization transformations used for the two models, in both cases we find four relevant tricritical eigenvalues which are essentially identical for both models. For the antiferromagnet, there is no indication of any decomposition of the tricritical point as predicted by mean-field theory, for ratios of intrasublattice to intersublattice coupling as small as (1/8).

Physical review. B, Condensed matter•Jun 1, 1986•62 citations

Multicritical behavior of the antiferromagnetic spin-3/2 Blume-Capel model: Finite-size-scaling and Monte Carlo studies

Smaîne Bekhechi, A. Benyoussef

Transfer-matrix finite-size-scaling calculations and Monte Carlo simulations are used to investigate the two-dimensional spin-3/2 Ising antiferromagnet in the presence of an external magnetic field and a single-ion potential. Comparison is made between the results of this study and previous mean-field calculations. The phase diagrams and the critical behavior of the model are discussed. In contrast to the mean-field picture, no decomposition of the tricritical point is observed.

Physical review. B, Condensed matter•Dec 1, 1997•48 citations

Stability of the tricritical point in a three-dimensional next-nearest-neighbor Ising antiferromagnet: A Monte Carlo simulation

Hans Jürgen Herrmann, David Paul Landau

We investigate via Monte Carlo simulations the multicritical behavior of Ising antiferromagnets in three dimensions. Even for ratios between the intrasublattice and intersublattice couplings as small as R=0.05, we find tricritical behavior and no evidence for a decomposition into a critical end point and a double critical end point as predicted by mean-field theories.

Physical review. B, Condensed matter•Jul 1, 1993•34 citations

Tricriticality and finite-size scaling in the triangular Blume-Capel ferromagnet

Dimitrios Mataragkas, Alexandros Vasilopoulos, Nikolaos G. Fytas +1 more

We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.

arXiv:2411.11689•Physical Review Research•Feb 27, 2025•13 citations

Triangular Ising model with nearest- and next-nearest-neighbor couplings in a field

Xiaofeng Qian, Henk W. J. Blöte

The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.

arXiv:physics/0405055•Physical Review E•Sep 21, 2004•20 citations

Finite-size scaling study of the two-dimensional Blume-Capel model

Paul D. Beale

The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (${T}_{t}$=0.610\ifmmode\pm\else\textpm\fi{}0.005, ${D}_{t}$=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.

Physical review. B, Condensed matter•Feb 1, 1986•119 citations
We study the two-dimensional, antiferromagnetic Blume-Capel model on a square lattice by numerical transfer-matrix and Monte Carlo finite-size scaling methods, in order to investigate whether the line of tricritical points in this model may be decomposed into lines of critical end points and double critical points, as predicted by mean-field theory and recent Monte Carlo simulations on a three-dimensional cubic lattice. Conclusive numerical evidence is obtained, indicating that such decomposition does u/Inot occur in this two-dimensional model. The nondecomposition is explained in terms of the large fluctuations in the two-dimensional nearest-neighbor model, and we speculate that the situation may be different in two-dimensional antiferromagnetic models with weak, ferromagnetic next-nearest-neighbor interactions.
Using a Monte Carlo renormalization-group (MCRG) method we study tricritical behavior in two very different two-dimensional spin models: the Blume-Capel ferromagnet and the next-nearest-neighbor Ising antiferromagnet. The MCRG method is used to locate accurately tricritical points independent of the convergence of eigenvalue estimates. Despite the different symmetries and the different renormalization transformations used for the two models, in both cases we find four relevant tricritical eigenvalues which are essentially identical for both models. For the antiferromagnet, there is no indication of any decomposition of the tricritical point as predicted by mean-field theory, for ratios of intrasublattice to intersublattice coupling as small as (1/8).
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.
The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (Tt{T}_{t}Tt​=0.610\ifmmode\pm\else\textpm\fi{}0.005, Dt{D}_{t}Dt​=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.