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

Abstract

Renormalization of the spin-wave spectrum is discussed in a cubic ferromagnet with dipolar forces at ${T}_{C}⪢T\ensuremath{\geqslant}0$. The first $1∕S$ corrections to the bare spectrum ${ϵ}_{\mathbf{k}}=\sqrt{D{k}^{2}(D{k}^{2}+S{\ensuremath{\omega}}_{0}\phantom{\rule{0.2em}{0ex}}{\mathrm{sin}}^{2}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{\mathbf{k}})}$ are considered in detail, where $D$ is the spin-wave stiffness, ${\ensuremath{\theta}}_{\mathbf{k}}$ is the angle between $\mathbf{k}$ and the magnetization, and ${\ensuremath{\omega}}_{0}$ is the characteristic dipolar energy. In accordance with previous results we obtain the thermal renormalization of constants $D$ and ${\ensuremath{\omega}}_{0}$ in the expression for the bare spectrum. In addition, a number of previously unknown features are revealed. We observe terms that depend on the azimuthal angle of the momentum $\mathbf{k}$. They are proportional to the square invariant $\ensuremath{\mid}{k}_{x}^{2}\ensuremath{-}{k}_{y}^{2}\ensuremath{\mid}∕k$ in the case of a simple cubic lattice and to the triangular invariant $\mathrm{Im}[{({k}_{x}+i{k}_{y})}^{3}]∕{k}^{2}$ in face- and body-centered cubic lattices, where we assume that the $z$ axis is directed along the magnetization. An isotropic term proportional to $k$ is obtained which makes the spectrum linear rather than quadratic when $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{\mathbf{k}}=0$ and $k⪡{\ensuremath{\omega}}_{0}∕D$. In particular a spin-wave gap proportional to $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{\mathbf{k}}$ is observed. Essentially, the thermal contributions 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 the 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 $T⪡{\ensuremath{\omega}}_{0}$. However, analysis of the entire perturbation series is still required to derive the spectrum and LDSS when $T⪢{\ensuremath{\omega}}_{0}$. Such an analysis is out of the scope of the present paper.