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Physical Review

Application of the Cluster Variation Method to the Heisenberg Model with Arbitrary Spin and Range of Exchange

Tohru Morita, Tomoyasu Tanaka

Published May 6, 1966• 65 citations
Publisher

Abstract

The cluster variation method for the cooperative phenomena proposed by Kikuchi and reformulated and generalized by Morita, is applied to the Heisenberg model with arbitrary spin and range of exchange. A general expression for the two-body reduced density matrix is obtained in the approximation in which the clusters of pairs of lattice sites are retained correctly. The constant-coupling approximation for the Heisenberg model of $S\ensuremath{\geqq}1$ is shown to be derived by satisfying the reducibility conditions ${{\mathrm{tr}}_{k\ensuremath{\rho}}}^{(2)}(j, k)={\ensuremath{\rho}}^{(1)}(j)$ only partly, requiring the consistency for the zeroth and first moments of ${S}_{\mathrm{jz}}$ and ignoring the consistency for the second to $2S\mathrm{th}$ moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.

DOI
10.1103/physrev.145.288

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The cluster variation method for the cooperative phenomena proposed by Kikuchi and reformulated and generalized by Morita, is applied to the Heisenberg model with arbitrary spin and range of exchange. A general expression for the two-body reduced density matrix is obtained in the approximation in which the clusters of pairs of lattice sites are retained correctly. The constant-coupling approximation for the Heisenberg model of S\ensuremath≧1S\ensuremath{\geqq}1S\ensuremath≧1 is shown to be derived by satisfying the reducibility conditions trk\ensuremathρ(2)(j,k)=\ensuremathρ(1)(j){{\mathrm{tr}}_{k\ensuremath{\rho}}}^{(2)}(j, k)={\ensuremath{\rho}}^{(1)}(j)trk\ensuremathρ​(2)(j,k)=\ensuremathρ(1)(j) only partly, requiring the consistency for the zeroth and first moments of Sjz{S}_{\mathrm{jz}}Sjz​ and ignoring the consistency for the second to 2Sth2S\mathrm{th}2Sth moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.
A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.
In this thesis we present a novel variational cluster approximation for Heisenberg spin systems. It is based on the self-energy functional theory by Potthoff for fermionic and bosonic models with local interactions. To develop a similar method for spin systems, we derive a free energy functional which is the starting point of the approximation. Within this approximation, we find an analytical expression to evaluate the free energy by tiling the real system into a set of clusters. We describe the technical details of the spin variational cluster approximation and the evaluation of the free energy. The method is tested for the antiferromagnetic spin-1/2 Heisenberg chain with first- and second-neighbour interactions. Thereby, we investigate the dependence of the approximation on cluster size and the choice of variational parameters. The opportunities and limitations as well as future applications of the method are thoroughly discussed.
The coupled-cluster method (CCM) has already been successfully applied by us to anisotropic quantum antiferromagnetic models (XXZ model) with spin s=1/2 in one and two dimensions. This approach is now generalized to the same models but with spin s>1/2. We mainly consider the one-dimensional system in this article. Several systematic approximation schemes developed previously are employed. The ground-state energy is obtained as a function of the spin s and anisotropy. It is found that, for a given approximation that contains long-range contributions, the critical value of the anisotropy for the Heisenberg-Ising phase transition rapidly approaches the classical value of 1 as the spin s increases. We also generalize one of the schemes to a hypercubic lattice of arbitrary dimensionality. By making a detailed comparison with spin-wave theory, perturbation theory, and other variational approaches, some additional intrinsic features of the CCM treatments are emphasized.
The method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin Sz=±1S^z=\pm1Sz=±1) the expressions for pair and ternary correlation functions within two-particle approximation in { q }\set{q}{q}-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility χ({ q },E)χ(\set{q},E)χ({q},E) is found. In the 1D case and in the absence of external field the expression for χ({ q },E)χ(\set{q},E)χ({q},E) is exact. For the Emery-Blume-Griffiths model (Sz=−2,0,2S^z=-2,0,2Sz=−2,0,2) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility χ({ q })χ(\set{q})χ({q}) of KD2PO4KD_2PO_4KD2​PO4​ ferroelectrics.
The cluster variation method, in its recent formulation using the Moebius inversion, is applied to the study of the critical properties of the spin-1 BlumeCapel model. In order to stress the various steps of the approximation a face-centered-cubic lattice is considered and the free energy of the system is written in several degrees of approximations: single site (one-body), pair (two-body), triangle (three-body), tetrahedron (four-body). The case of the pair approximation is discussed in detail and the resulting phase diagram is compared with that obtained using other methods. The behavior of the dipolar and quadrupolar order parameters and of the pair-correlation functions is examined in three typical cases: second order transition, first order transition, no transitions.
In this paper we have applied the coupled-cluster method to investigate the ground state of the spin-1 Heisenberg antiferromagnet with anisotropic exchange interaction and easy-plane single-ion anisotropy. A coupled-cluster approximation up to the second level has been performed to evaluate the ground-state energy and staggered magnetization for various lattices such as linear chain, honeycomb, square, simple cubic, and body-centered-cubic lattice. The convergence of our results is fairly rapid, and thus the coupled-cluster method seems to be a practical tool for calculating ground-state energy of a spin system since only the first few levels of approximation will be needed in actual practice.