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Systematic approach to critical phenomena by the extended variational method and coherent-anomaly method

Naoki Kawashima, Makoto Katori, Constantino Tsallis, Masuookyo Suzuki

Published Jan 1, 1989• 0 citations
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Abstract

A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.

Related papers

SYSTEMATIC APPROACH TO CRITICAL PHENOMENA BY THE EXTENDED VARIATIONAL METHOD AND COHERENT-ANOMALY METHOD

Naoki Kawashima, Makoto Katori, Constantino Tsallis +1 more

A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.

International Journal of Modern Physics B•Jun 1, 1990•4 citations

On the CAM Canonicality of the Cluster-Variation Approximations

Makoto Katori, Masuo Suzuki

The cluster-variation method (CVM) proposed by Kikuchi is a general theory to give approximations, which are useful to discuss phase transitions qualitatively in many systems. Recently one of the present authors (M. S.) proposed the coherent-anomaly method (CAM). If we have a well-behaved series of approximations, which is called a canonical series, we can estimate critical exponents following the CAM. In this paper it is demonstrated by using the ferromagnetic Ising models that the CVM will provide a canonical series which shows coherent anomaly. Our result implies that combining the CVM with the CAM will give a powerful method to study phase transitions and critical phenomena.

Progress of Theoretical Physics Supplement•Jan 1, 1994•10 citations

Coherent Anomaly Method in Critical Phenomena. I.

Masuo Suzuki, Makoto Katori, Xiao Hu

A coherent anomaly method (CAM) is formulated for critical phenomena on the basis of the general idea of CAM for cooperative phenomena proposed by Suzuki. The coherent anomalies of response functions are derived phenomenologically using Kubo's linear response theory and Fisher's scaling form of correlation functions. This yields the relation between the coherent anomaly exponents and non-classical critical exponents. The convergence of the CAM critical points is proven rigorously. Some explicit applications of the CAM theory based on systematic cluster-mean-field approximations and on the mean-field transfer-matrix method are presented to show how useful the CAM is in studying critical phenomena. The CAM theory of critical dynamics is also formulated to give the dynamical critical exponent Δ ≃1.84 in the two-dimensional kinetic Ising model.

Journal of the Physical Society of Japan•Sep 1, 1987•113 citations

Accurate estimates of 3D Ising critical exponents using the Coherent-Anomaly Method

Miroslav Kolesik, Masuo Suzuki

An analysis of the critical behavior of the three-dimensional Ising model using the coherent-anomaly method (CAM) is presented. Various sources of errors in CAM estimates of critical exponents are discussed, and an improved scheme for the CAM data analysis is tested. Using a set of mean-field type approximations based on the variational series expansion approach, accuracy comparable to the most precise conventional methods has been achieved. Our results for the critical exponents are given by α = 0.108(5), β = 0.327(4), γ = 1.237(4) and δ = 4.77(5).

•Jan 1, 2008•12 citations

Study of Critical Phenomena by Spin-Cluster Approximation Series: Ising Spin System

Hiroshi Mano, Katsumi Nakao

A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.

Journal of the Physical Society of Japan•Feb 15, 1991•6 citations

Coherent Anomaly Method in Critical Phenomena. II. Applications to the Two- and Three-Dimensional Ising Models

Makoto Katori, Masuo Suzuki

A systematic procedure of constructing a series of cluster-mean-field approximations is established to investigate the coherent anomalies of the mean-field critical coefficients \barχ( L , T c ( L )) in the two- and three-dimensional Ising models. The values of critical exponents as well as the critical temperatures are estimated by means of the new method (Coherent Anomaly Method) as T c * =2.230(0.038) J / k B , γ=1.785(0.116) for the two-dimensional square lattice and T c * =4.529(0.058) J / k B , γ=1.258(0.068) for the three-dimensional simple cubic lattice. The results suggest that the present approach to study critical phenomena is very promising.

Journal of the Physical Society of Japan•Sep 1, 1987•73 citations
A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.
A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.
The cluster-variation method (CVM) proposed by Kikuchi is a general theory to give approximations, which are useful to discuss phase transitions qualitatively in many systems. Recently one of the present authors (M. S.) proposed the coherent-anomaly method (CAM). If we have a well-behaved series of approximations, which is called a canonical series, we can estimate critical exponents following the CAM. In this paper it is demonstrated by using the ferromagnetic Ising models that the CVM will provide a canonical series which shows coherent anomaly. Our result implies that combining the CVM with the CAM will give a powerful method to study phase transitions and critical phenomena.
A coherent anomaly method (CAM) is formulated for critical phenomena on the basis of the general idea of CAM for cooperative phenomena proposed by Suzuki. The coherent anomalies of response functions are derived phenomenologically using Kubo's linear response theory and Fisher's scaling form of correlation functions. This yields the relation between the coherent anomaly exponents and non-classical critical exponents. The convergence of the CAM critical points is proven rigorously. Some explicit applications of the CAM theory based on systematic cluster-mean-field approximations and on the mean-field transfer-matrix method are presented to show how useful the CAM is in studying critical phenomena. The CAM theory of critical dynamics is also formulated to give the dynamical critical exponent Δ ≃1.84 in the two-dimensional kinetic Ising model.
An analysis of the critical behavior of the three-dimensional Ising model using the coherent-anomaly method (CAM) is presented. Various sources of errors in CAM estimates of critical exponents are discussed, and an improved scheme for the CAM data analysis is tested. Using a set of mean-field type approximations based on the variational series expansion approach, accuracy comparable to the most precise conventional methods has been achieved. Our results for the critical exponents are given by α = 0.108(5), β = 0.327(4), γ = 1.237(4) and δ = 4.77(5).
A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.
A systematic procedure of constructing a series of cluster-mean-field approximations is established to investigate the coherent anomalies of the mean-field critical coefficients \barχ( L , T c ( L )) in the two- and three-dimensional Ising models. The values of critical exponents as well as the critical temperatures are estimated by means of the new method (Coherent Anomaly Method) as T c * =2.230(0.038) J / k B , γ=1.785(0.116) for the two-dimensional square lattice and T c * =4.529(0.058) J / k B , γ=1.258(0.068) for the three-dimensional simple cubic lattice. The results suggest that the present approach to study critical phenomena is very promising.