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Journal de Physique I

New variational series expansions for lattice models

M. Kolesík, Ladislav Šamaj

Published Jan 1, 1993• 9 citations
Publisher

Abstract

For the symmetric two-state vertex model on the honeycomb lattice we construct a series expansion of the free energy which, at finite order, depends on free gauge parameters. We treat these gauge parameters as variational ones, and derive a canonical series of classical approximations which possesses the property of coherent anomaly. As a test model we use the vertex formulation of the Ising antiferromagnet in a field and, within the coherent-anomaly method, determine with a high accuracy its critical frontier and exponent γ. Numerical checks on the constancy of critical exponents along the phase boundary are presented, too.

DOI
10.1051/jp1:1993119

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For the symmetric two-state vertex model on the honeycomb lattice we construct a series expansion of the free energy which, at finite order, depends on free gauge parameters. We treat these gauge parameters as variational ones, and derive a canonical series of classical approximations which possesses the property of coherent anomaly. As a test model we use the vertex formulation of the Ising antiferromagnet in a field and, within the coherent-anomaly method, determine with a high accuracy its critical frontier and exponent γ. Numerical checks on the constancy of critical exponents along the phase boundary are presented, too.
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.
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 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.
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).