Tensor of the inhomogeneous dynamic susceptibility of an anisotropic Heisenberg ferromagnet and Bogolyubov inequalities. I. Single-particle matrix Green's function and transverse components of the susceptibility tensor

Two-time thermal Green's functions are used to consider the tensor of the inhomogeneous dynamic susceptibility $\chi^{\alpha\beta}(k,E)$ of the generalized anisotropic three-dimensional Heisenberg model with spin $1/2$. The transverse components ($\alpha,\beta=x,y$) are obtained by means of the single-particle matrix Green's function in the generalized Hartree–Fock approximation. The Tyablikov approximation is used to analyze the dependence of the diagonal components $\chi^x_k$ and $\chi^y_k$ in the static limit $E=0$ on the quasimomentum, the anisotropy, and the external field in a wide range of temperatures. It is shown in particular that for degenerate models of the easy plane type (including the isotropic case) which in the absence of a field have a gapless spectrum one or both of the components $\chi^x_k$ and $\chi^y_k$ diverge at $k=0$ in the ferromagnetic region, while in the paramagnetic region they have the Ornstein–Zernike form. The obtained results are in agreement with Bogolyubov's rigorous inequalities, which are generalized to the case of arbitrary anisotropy.