Renormalization of the spin-wave spectrum in three-dimentional ferromagnets with dipolar interaction

Renormalization of the spin-wave spectrum is discussed in a cubic ferromagnet with dipolar forces at $T_C\gg T\ge 0$. First 1/S-corrections are considered in detail to the bare spectrum $\epsilon_{\bf k} = \sqrt{Dk^2 (Dk^2 + S\omega_0\sin^2\theta_{\bf k})}$,where$D$isthespin-wavestiffness,$\theta_{\bf k}$istheanglebetween$\bf k$andthemagnetizationand$\omega_0$ is the characteristic dipolar energy. In accordance with previous results we obtain the thermal renormalization of constants $D$ and ${\omega }_0$ in the expression for the bare spectrum. Besides, a number of previously unknown features are revealed. We observe terms which depend on azimuthal angle of the momentum $\mathbf{k}$. It is obtained an isotropic term proportional to $k$ which makes the spectrum linear rather than quadratic when $\sin\theta_{\bf k}=0$and$k \ll \omega_0/D$\mathrm{.}Inparticularaspin-wavegapproportionalto$\sin\theta_{\bf k}$ is observed. Essentially, thermal contribution from the Hartree-Fock diagram to the isotropic correction as well as to the spin-wave gap are proportional to the demagnetizing factor in the direction of domain magnetization. This nontrivial behavior is attributed to the long-range nature of the dipolar interaction. It is shown that the gap screens infrared singularities of the first 1/S-corrections to the spin-wave stiffness and longitudinal dynamical spin susceptibility (LDSS) obtained before. We demonstrate that higher order 1/S-corrections to these quantities are small at $T\ll {\omega }_0$. However the analysis of the entire perturbation series is still required to derive the spectrum and LDSS when $T\gg {\omega }_0$.