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Journal of the Physical Society of Japan

Coherent-Anomaly Method Applied to the Lattice Obtained by the Dual Transformation in the Ising Model

Takehiko Oguchi, Hidetsugu Kitatani

Published Sep 15, 1989• 4 citations
Publisher

Abstract

The specific heats at the critical temperatures of the dual lattice of the Ising model on a square lattice were calculated by the canonical series of cluster approximations. Then the specific heats of the original lattice were calculated through the dual transformation. Finally the critical exponent α' below the transition temperature was determined as α'=0 by the coherent-anomaly method.

DOI
10.1143/jpsj.58.3033

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The specific heats at the critical temperatures of the dual lattice of the Ising model on a square lattice were calculated by the canonical series of cluster approximations. Then the specific heats of the original lattice were calculated through the dual transformation. Finally the critical exponent α' below the transition temperature was determined as α'=0 by the coherent-anomaly method.
The coherent-anomaly method proposed by one of the authors (M.S.) combined with a transfer-matrix version of the double-cluster approximation is applied to antiferromagnetic Ising models. We determined both the critical temperature and the critical exponent γ as functions of an external magnetic field. For a square lattice, our results are in a good agreement with the ones obtained recently by the phenomenological renormalization method. For triangle and simple cubic lattices, our results are less definite but still in a qualitative agreement with values predicted by other methods.
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.
On the basis of the coherent-anomaly method (CAM) proposed by Suzuki, the static and dynamic critical exponents of the Ising model in the triangle and the square lattice are calculated using the cluster variation method (CVM) and the path probability method (PPM) devised by Kikuchi within analytical tractable approximations. The values of critical temperatures and critical exponents estimated from three approximations with different levels are generally reasonable. The two-dimensional results are also compared with those of the simple cubic lattice.
The critical exponent ν of correlation length is calculated on the basis of the coherent anomaly method (CAM) by using the pair, the cactus-square and the square approximations of the cluster variation method (CVM) in the D (≥2)-dimensional cubic lattice of the Ising model. These simple analytical approximations give good values of critical exponent ν and critical temperature in the two- and three-dimensional lattices. However, further consideration is needed to determine a critical dimension in the D -dimensional Ising model by the CAM.
The coherent-anomaly scaling is investigated for the approximations derived on the basis of the cluster-variation method (CVM). All of the mean-field critical coefficients are calculated in the Weiss, the Bethe and the Kikuchi approximations for the three-dimensional Ising model on the simple cubic lattice. The dynamical coefficients of the relaxation time are also obtained in the corresponding approximations for the kinetic Ising model. The non-classical values of critical exponents as well as the true value of critical temperature are estimated by extrapolating these approximants by the coherent-anomaly method. The results suggest that the approximations by the CVM possess the property of coherent-anomaly scaling.