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Journal of the Physical Society of Japan

Coherent-Anomaly Analysis of Series Expansions and Its Application to the Ising Model

Yosinori Kinosita, Naoki Kawashima, Masuo Suzuki

Published Nov 15, 1992• 8 citations
Publisher

Abstract

The applicability of the coherent-anomaly method (CAM) to series expansions is checked using test functions and the critical exponents γ for the Ising models are estimated by this method. Among three canonical series studied here, the Padé series and the continued-fraction series are most effective in their convergence and the accuracy of estimation. There are some functions which are difficult to analyze with the power-series CAM. Our estimate for γ of the three-dimensional Ising model is 1.246. This value is higher than the latest results obtained by means of other methods taking into account the correction-to-scaling term.

DOI
10.1143/jpsj.61.3887

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The applicability of the coherent-anomaly method (CAM) to series expansions is checked using test functions and the critical exponents γ for the Ising models are estimated by this method. Among three canonical series studied here, the Padé series and the continued-fraction series are most effective in their convergence and the accuracy of estimation. There are some functions which are difficult to analyze with the power-series CAM. Our estimate for γ of the three-dimensional Ising model is 1.246. This value is higher than the latest results obtained by means of other methods taking into account the correction-to-scaling term.
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 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.
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.
We propose and test a new method for generating canonical sequences for analysis by the coherent anomaly method (CAM) from non-mean-field approximations. By intentionally underestimating the rate of convergence of exact-diagonalization values for the mass or energy gaps of finite systems, we form families of sequences of gap estimates. The gap estimates cross zero with generically nonzero linear terms in their Taylor expansions, so that for each member of these sequences of estimates. Thus, the CAM can be used to determine . Our freedom in deciding exactly how to underestimate the convergence allows us to choose the sequence that displays the clearest coherent anomaly. We demonstrate this approach on the two-dimensional ferromagnetic Ising model, for which . We also use it on the three-dimensional ferromagnetic Ising model, finding , in good agreement with other estimates. Finally, we apply it to an antiferromagnetic spin-1 Heisenberg chain, finding at the phase transition between the Haldane phase and the dimerized phase, in agreement with the field-theoretic prediction . Although the specific systems used to test the extrapolation-CAM procedure involve finite system sizes, the method could be applied to other finite approximations, such as systematic variational approximations.
The coherent-anomaly method (CAM) is applied to the kinetic Ising model. Dynamical cluster-mean-field approximations are formulated to obtain a series of the dynamical mean-field critical coefficients. The coherent-anomaly scaling relations are derived on the basis of the scaling form of the generating function in nonequilibrium systems to estimate the exponent Δ of the critical slowing down. The dynamical critical exponent is estimated as Δ⋍2.15(±0.02){\varDelta}{\backsimeq}2.15({\pm}0.02)Δ⋍2.15(±0.02) for the kinetic Ising model on the two-dimensional triangular lattice.