Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbor triangular XXZ model within the $1/S$ expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy $J_{xy}/J_{zz}\ll 1$. We show that, in the limit in which $1/S\to 0$ and $J_{xy}\to 0$ at fixed $V=J_{zz}/\left(S{J}_{xy}\right)$, the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin wave ($V\to 0$) and a strong-coupling limit ($V\to \infty$) and encodes in a simple framework the regimes discussed by Kleine et al. [Z. Phys. B: Condens. Matter 86, 405 (1992); Z. Phys. B: Condens. Matter, 87, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which can be traced to a simple symmetry of the classical energy. The model thus offers a transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order. In the calculation, we introduce a self-consistent renormalization of the energy scale and of the pseudo-Goldstone energy gap; the latter renormalization is essential to remove infrared divergences in the on-shell corrections to the energy dispersion. Finally, we discuss qualitatively the structure of the one-loop corrections in comparison with the spectrum observed experimentally in ${\mathrm{K}}_2\mathrm{Co}{\left({\mathrm{SeO}}_3\right)}_2$.

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