Tensor of the inhomogeneous dynamic susceptibility of an anisotropic Heisenberg ferromagnet and Bogolyubov inequalities

The method of two-time thermal Green's functions is used to consider the tensor of the inhornogeneous dynamic susceptibility $\chi^{\alpha\beta}(k,E)$ of the generalized anisotropic Heisenberg model with spin $1/2$. The longitudinal component ($\alpha=\beta=z$) is obtained by means of the collective matrix Green's function in the random phase approximation by means of the single-particle dynamics in the Tyablikov approximation. It is shown that these approximations are consistent at $E=0$, since for some models they ensure fulfillment of the symmetry conditions and sum rules for the longitudinal and transverse binary spin correlation functions in the paramagnetic temperature range. An analysis is made of the asymptotic behavior of $\chi^{zz}(k,0)$ with respect to the quasimomentum, the anisotropy, and the external field in a wide range of temperatures. It is shown that for degenerate models such as the easy plane model and the isotropic model, which have a gapless single-particle spectrum in the absence of an external magnetic field, $\chi^{zz}(k,0)$ diverges at $k=0$ in the ferromagnetic region and at the Curie point, while in the paramegnetic region it has the Ornstein–Zernike form. The obtained results agree with Bogolyubov's rigorous inequality applied to estimate $\chi^{zz}(k,0)$.