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Extrapolation-CAM theory for critical exponents

Richards, H. L., Hatano, N., M. A. Novotny

Published Jan 1, 1997• 0 citations
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Abstract

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 $\nu = 1$ for each member of these sequences of estimates. Thus, the Coherent Anomaly Method can be used to determine $\nu$. 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 $\nu = 1$. We also use it on the three-dimensional ferromagnetic Ising model, finding $\nu \approx 0.629$, in good agreement with other estimates.

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Extrapolation-CAM theory for critical exponents

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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.

arXiv:cond-mat/9703222•Journal of Physics A Mathematical and General•Sep 21, 1997•0 citations

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Coherent-anomaly method in zero-temperature phase transitions in quantum spin systems

Yoshihiko Nonomura, Masuo Suzuki

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.

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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.
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 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 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.
A coherent anomaly method (CAM) is formulated for critical phenomena on the basis of the general idea of CAM for cooperative phenomena proposed by Suzuki. The coherent anomalies of response functions are derived phenomenologically using Kubo's linear response theory and Fisher's scaling form of correlation functions. This yields the relation between the coherent anomaly exponents and non-classical critical exponents. The convergence of the CAM critical points is proven rigorously. Some explicit applications of the CAM theory based on systematic cluster-mean-field approximations and on the mean-field transfer-matrix method are presented to show how useful the CAM is in studying critical phenomena. The CAM theory of critical dynamics is also formulated to give the dynamical critical exponent Δ ≃1.84 in the two-dimensional kinetic Ising model.
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.