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Journal of Physics A Mathematical and General

An approach to critical phenomena from one subsystem based on the CAM analysis

Xiao Hu, Masuo Suzuki

Published Jul 7, 1990• 12 citations
Publisher

Abstract

Two systematic series of multi-body effective field approximations are constructed and the CAM analyses are performed for the Ising mode on the two-dimensional square lattice. In the first series a two-body effective field is introduced in addition to the usual one-body one and a set of strips are used. The critical temperature and the critical index of the susceptibility are estimated as T* c approximately=2.2697 J/k B and gamma approximately=1.749. In the second series only a three-line strip is used and applied with increasing numbers of multi-body effective fields. The expected CAM scaling behaviour has also been observed.

DOI
10.1088/0305-4470/23/13/039

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Two systematic series of multi-body effective field approximations are constructed and the CAM analyses are performed for the Ising mode on the two-dimensional square lattice. In the first series a two-body effective field is introduced in addition to the usual one-body one and a set of strips are used. The critical temperature and the critical index of the susceptibility are estimated as T* c approximately=2.2697 J/k B and gamma approximately=1.749. In the second series only a three-line strip is used and applied with increasing numbers of multi-body effective fields. The expected CAM scaling behaviour has also been observed.
The double-cluster approximation (DCA) and the multi-effective-field approximation (MEFA) are formulated for the two-dimensional S=1/2 transverse Ising model on a square lattice in the ground state. These two cluster-effective-field approximations expand the sizes of clusters effectively. Combining some suitable series of these approximations with the coherent-anomaly method (CAM), the authors have analysed the critical phenomenon of this model. They have obtained g * c =1.528+or-0.016 and gamma =1.25+or-0.08 using the DCA, and gamma =1.26+or-0.10 using the MEFA. These estimates are consistent with the other results, g * c approximately=1.52 and gamma approximately=1.24. Their scheme is expected to be useful in the study of frustrated systems, because various shapes of clusters can be used in their calculations.
The systematic Weiss-like and Bethe-like approximations based on the mean-field transfer-matrix method are used to investigate the asymptotic behavior of the induced magnetization on a semi-infinite square lattice, and to investigate the wave-number dependence of the susceptibility in a nonuniform external field. The critical exponents ν, ν', η i and η are estimated following the general CAM prescription. A new scaling relation ν·η i =γ is obtained in the framework of the finite-degree-of-approximation scaling. Together with previous papers, all the static critical exponents have been estimated by the CAM, and are shown to satisfy the well-known scaling relations.
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.
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.
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).