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

International Journal of Modern Physics B

THREE- AND FOUR-BODY CORRELATION FUNCTIONS FOR THE BLUME-CAPEL MODEL

Carla Buzano, L. R. Evangelista

Published Feb 28, 1993• 2 citations
Publisher

Abstract

The Cluster Variation Method (CVM), in its recent formulation using the Moebius Inversion, is applied to study the critical properties of the face-centered-cubic Blume-Capel model in the triangle and tetrahedron approximations. We develop a procedure which preserves the characteristics of Kikuchi’s Natural Iteration Method, moreover reducing considerably the number of involved variables. The resulting phase diagram is compared with that obtained from lower orders of CVM approximation (single-site and pair) and from other methods (series expansions, Monte Carlo, …). The tetrahedron approximation shows a good agreement with these high-precision methods, in particular for the location of the tricritical point. The behavior of order parameters and multisite (up to four-body) correlation functions is determined for all significant regions of the phase space.

DOI
10.1142/s0217979293002316

Related papers

Cluster variation method in the analysis of spin-1 models

Carla Buzano, L. R. Evangelista

The cluster variation method, in its recent formulation using the Moebius inversion, is applied to the study of the critical properties of the spin-1 BlumeCapel model. In order to stress the various steps of the approximation a face-centered-cubic lattice is considered and the free energy of the system is written in several degrees of approximations: single site (one-body), pair (two-body), triangle (three-body), tetrahedron (four-body). The case of the pair approximation is discussed in detail and the resulting phase diagram is compared with that obtained using other methods. The behavior of the dipolar and quadrupolar order parameters and of the pair-correlation functions is examined in three typical cases: second order transition, first order transition, no transitions.

•Jan 1, 1991•0 citations

Cluster-variation–Padé-approximants method and the critical exponents of the fcc Ising model

Alessandro Pelizzola

We report results of the application of a recently proposed technique, which combines the cluster variation method (CVM) and Pad\'e approximants, to the analysis of the critical behavior of the face-centered-cubic Ising model. The highest-order CVM approximation ever considered for this lattice is used to estimate magnetization and susceptibility, which are then analyzed by means of Dlog Pad\'e approximants in order to extract information about the critical behavior, taking also into account the effects of confluent singularities. Our results are in good agreement with the most recent simulation and series expansion estimates and require a remarkably small numerical effort. \textcopyright{} 1996 The American Physical Society.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1996•14 citations

Linked-cluster series analysis of the Blume-Capel model

Yung‐Li Wang, Felix Lee

Linked-cluster series have been obtained for the Blume-Capel model in an fcc lattice to the eighth order. We show that the linked-cluster series aided by the standard extrapolation techniques provide an effective method in the study of models which display first-order phase transitions as well as second-order phase transitions. The full phase diagram and the tricritical phenomena have been studied. Results are compared with those of the Monte Carlo simulation and the high-temperature and low-temperature series analysis.

Physical review. B, Condensed matter•May 1, 1984•16 citations

Critical and tricritical behavior of the d = 3 Blume-Capel model: Results from small-scale Monte Carlo simulations

Leïla Moueddene, Nikolaos G. Fytas, Bertrand Berche

We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.

arXiv:2410.01710•Physical review. E•Dec 23, 2024•12 citations

Renormalization-group results for the Blume-Capel model in two and three dimensions

Theodore W. Burkhardt, H. J. F. Knops

A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for $d=2,3$ dimensions.

Physical review. B, Solid state•Feb 1, 1977•64 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
The Cluster Variation Method (CVM), in its recent formulation using the Moebius Inversion, is applied to study the critical properties of the face-centered-cubic Blume-Capel model in the triangle and tetrahedron approximations. We develop a procedure which preserves the characteristics of Kikuchi’s Natural Iteration Method, moreover reducing considerably the number of involved variables. The resulting phase diagram is compared with that obtained from lower orders of CVM approximation (single-site and pair) and from other methods (series expansions, Monte Carlo, …). The tetrahedron approximation shows a good agreement with these high-precision methods, in particular for the location of the tricritical point. The behavior of order parameters and multisite (up to four-body) correlation functions is determined for all significant regions of the phase space.
The cluster variation method, in its recent formulation using the Moebius inversion, is applied to the study of the critical properties of the spin-1 BlumeCapel model. In order to stress the various steps of the approximation a face-centered-cubic lattice is considered and the free energy of the system is written in several degrees of approximations: single site (one-body), pair (two-body), triangle (three-body), tetrahedron (four-body). The case of the pair approximation is discussed in detail and the resulting phase diagram is compared with that obtained using other methods. The behavior of the dipolar and quadrupolar order parameters and of the pair-correlation functions is examined in three typical cases: second order transition, first order transition, no transitions.
We report results of the application of a recently proposed technique, which combines the cluster variation method (CVM) and Pad\'e approximants, to the analysis of the critical behavior of the face-centered-cubic Ising model. The highest-order CVM approximation ever considered for this lattice is used to estimate magnetization and susceptibility, which are then analyzed by means of Dlog Pad\'e approximants in order to extract information about the critical behavior, taking also into account the effects of confluent singularities. Our results are in good agreement with the most recent simulation and series expansion estimates and require a remarkably small numerical effort. \textcopyright{} 1996 The American Physical Society.
Linked-cluster series have been obtained for the Blume-Capel model in an fcc lattice to the eighth order. We show that the linked-cluster series aided by the standard extrapolation techniques provide an effective method in the study of models which display first-order phase transitions as well as second-order phase transitions. The full phase diagram and the tricritical phenomena have been studied. Results are compared with those of the Monte Carlo simulation and the high-temperature and low-temperature series analysis.
We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.
A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for d=2,3d=2,3d=2,3 dimensions.
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.