Slow convergence of spin-wave expansion and magnon dispersion in the 1/3 plateau of the triangular XXZ antiferromagnet

Motivated by recent experiments on the quantum magnet ${\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}$, 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 $\ensuremath{\alpha}={J}_{xy}/{J}_{zz}$ instead of $1/S$ and calculate the magnon dispersion for any value of the spin $S$ at second order in $\ensuremath{\alpha}$, 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 ${\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}$, for which $\ensuremath{\alpha}\ensuremath{\simeq}0.08$.