Analytic description of spin waves in dipolar/octupolar pyrochlore magnets

We derive analytic forms for spin waves in pyrochlore magnets with dipolar-octupolar interactions, such as ${\mathrm{N}\mathrm{d}}_2{\mathrm{Z}\mathrm{r}}_2{\mathrm{O}}_7$. We obtain full knowledge of the diagonalized magnonic Hamiltonian within the linear spin wave approximation. We also consider the effect of a “breathing mode” as a perturbation of this system. The breathing mode lifts the degeneracy of the upper band of the spin wave dispersion along the direction $X\to W$ in $k$-space.