Exact structure of equations for the longitudinal correlation function in the anisotropic Heisenberg ferromagnet

A method developed to calculate equilibrium correlation functions is used to find the exact structure of the equations for the longitudinal correlation function and its spectrum for arbitrary anisotropy parameters in the Heisenberg model. A new decoupling scheme based on allowance for the semi-invariants of the correlation function is proposed; it follows from this scheme that allowance for only the first moment leads to the generalized Hartree–Fock approximation for strongly interacting systems. In the framework of this generalization and with allowance for the interaction of nearest neighbors, a closed system of equations is obtained for $\langle S_{\mathbf k}^zS_{-{\mathbf k}}^z\rangle$ and its spectrum. The connection between the results obtained here and the calculations of other authors is discussed.