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arXiv:cond-mat/9703222 · Journal of Physics A Mathematical and General

Extrapolation-CAM theory for critical exponents

Howard L Richards, Naomichi Hatano, M. A. Novotny

Published Sep 21, 1997• 0 citations
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Abstract

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.

DOI
10.1088/0305-4470/30/18/013

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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.
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 ν=1\nu = 1ν=1 for each member of these sequences of estimates. Thus, the Coherent Anomaly Method can be used to determine ν\nuν. 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 ν=1\nu = 1ν=1. We also use it on the three-dimensional ferromagnetic Ising model, finding ν≈0.629\nu \approx 0.629ν≈0.629, in good agreement with other estimates.
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).
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 cluster-effective-field theory, called the double-cluster approximation, is formulated for quantum spin systems at T=0. Combining this approximation with the coherent-anomaly method (CAM), critical phenomena in such systems can be analysed. To test their theory, the authors have estimated the critical points and critical exponents of the one-dimensional S=1/2 transverse Ising model and the one-dimensional S=1/2 XZ model. The estimates are in good agreement with the exact solutions. These results show that the CAM approach is useful in treating zero-temperature phase transitions in quantum spin systems.
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.