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 K${}_2$Co(SeO${}_3$)${}_2$, we study theoretically the excitation spectrum of the nearest-neighbour 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 $\alpha =J_{xy}\mathrm{/}{J}_{zz}$ instead of $1\mathrm{/}S$ 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; (ii) compared to the linear spin-wave predictions, our magnon dispersion presents a much better agreement with experimental results on K${}_2$Co(SeO${}_3$)${}_2$, for which $\alpha \simeq 0.08$.