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

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 $S\geqq 1$ is shown to be derived by satisfying the reducibility conditions ${{\mathrm{tr}}_{k\rho }}^{\mathrm{}\left(2\right)\mathrm{}}(j, k)={\rho }^{\mathrm{}\left(1\right)\mathrm{}}\mathrm{}\left(j\right)\mathrm{}$ 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.