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Journal of the Physical Society of Japan

Coherent-Anomaly Method Applied to the Quantum Heisenberg Model

Takehiko Oguchi, Hidetsugu Kitatani

Published Nov 15, 1988• 24 citations
Publisher

Abstract

The coherent-anomaly method proposed by M. Suzuki is applied to the spin 1/2 ferromagnetic Heisenberg model to estimate the critical exponents. The effective Hamiltonian method to small-sized clusters for the simple cubic lattice, and the inverse temperature power series expansion of the susceptibility for the face centered cubic lattice are combined with the CAM. Both results show that the CAM is useful for the quantum spin systems.

DOI
10.1143/jpsj.57.3973

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The coherent-anomaly method proposed by M. Suzuki is applied to the spin 1/2 ferromagnetic Heisenberg model to estimate the critical exponents. The effective Hamiltonian method to small-sized clusters for the simple cubic lattice, and the inverse temperature power series expansion of the susceptibility for the face centered cubic lattice are combined with the CAM. Both results show that the CAM is useful for the quantum spin systems.
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.
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 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.
On the basis of the coherent-anomaly method (CAM) proposed by Suzuki, the static and dynamic critical exponents of the Ising model in the triangle and the square lattice are calculated using the cluster variation method (CVM) and the path probability method (PPM) devised by Kikuchi within analytical tractable approximations. The values of critical temperatures and critical exponents estimated from three approximations with different levels are generally reasonable. The two-dimensional results are also compared with those of the simple cubic lattice.
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.
The critical exponent ν of correlation length is calculated on the basis of the coherent anomaly method (CAM) by using the pair, the cactus-square and the square approximations of the cluster variation method (CVM) in the D (≥2)-dimensional cubic lattice of the Ising model. These simple analytical approximations give good values of critical exponent ν and critical temperature in the two- and three-dimensional lattices. However, further consideration is needed to determine a critical dimension in the D -dimensional Ising model by the CAM.