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International Journal of Modern Physics B

COHERENT-ANOMALY METHOD FOR QUANTUM SPIN SYSTEMS

Nobuyasu Ito, Masuo Suzuki

Published Feb 1, 1988• 19 citations
Publisher

Abstract

The coherent-anomaly method (CAM) is applied to the Heisenberg model to test the applicability of the CAM for quantum spin systems. The Weiss, Bethe and constant coupling approximations are tried for the Heisenberg model on the simple cubic lattice and estimate the critical exponents of the susceptibility and spontaneous magnetization using the CAM. The results show that the CAM is also powerful for quantum spin systems. The detailed results of the Bethe approximation of the spin-1/2 isotropic Heisenberg model are presented.

DOI
10.1142/s0217979288000020

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Unknown eBooks•Jan 1, 1989•0 citations
The coherent-anomaly method (CAM) is applied to the Heisenberg model to test the applicability of the CAM for quantum spin systems. The Weiss, Bethe and constant coupling approximations are tried for the Heisenberg model on the simple cubic lattice and estimate the critical exponents of the susceptibility and spontaneous magnetization using the CAM. The results show that the CAM is also powerful for quantum spin systems. The detailed results of the Bethe approximation of the spin-1/2 isotropic Heisenberg model are presented.
The critical temperatures Tc and the mean-field critical coefficients of the susceptibility χ are calculated for the S=1/2 Heisenberg-Ising model on the simple cubic lattice by the Kikuchi approximation and its simplified versions. χ̄ shows a coherent anomaly. It means that the coherent-anomaly method (CAM) is applicable to evaluate the true critical temperature T*c and the non-classical critical exponent γ for the susceptibility. As a result, the dependence of T*c on an anisotropy parameter η is given. It is shown that the behavior of γ versus η approaches a fiat line step by step as the corrections up to the higher order approximation are taken into account for 0≦η<1. It implies that the anisotropy parameter η of the Heisenberg-Ising model is irrelevant for critical exponents, if η is less than 1, as expected by the renormalization-group argument.
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.
A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.
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 general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.