Motivated by recent experiments on the quantum magnet ${\mathrm{K}}_2\mathrm{Co}{\left({\mathrm{SeO}}_3\right)}_2$, we study theoretically the excitation spectrum of the nearest-neighbor triangular XXZ model in the limit of strong easy-axis anisotropy, within the up-up-down (1/3-plateau) phase. We make an expansion in $\alpha =J_{xy}/J_{zz}$ instead of $1/S$ and calculate the magnon dispersion for any value of the spin $S$ at second order in $\alpha$, with two important conclusions: (i) the $1/S$ expansion converges very slowly for $S=1/2$, making spin-wave theory quantitatively inaccurate up to very large orders, and (ii) compared to the linear spin-wave predictions, our magnon dispersion presents a much better agreement with experimental results on ${\mathrm{K}}_2\mathrm{Co}{\left({\mathrm{SeO}}_3\right)}_2$, for which $\alpha \simeq 0.08$.

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• Frustrated magnetism
• Magnetic insulators
• Triangular lattice
• Perturbative methods
• Spin wave theory