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We use a combination of symmetry analysis, phenomenological modelling andfirst-principles density functional theory to explore the interplay between themagnetic ground state and the detailed atomic structure in the hexagonalrare-earth manganites. We find that the magnetic ordering is sensitive to abreathing mode distortion of the Mn and O ions in the $ab$ plane, which isdescribed by the K\textsubscript{1} mode of the high-symmetry structure. Ourdensity functional calculations of the magnetic interactions indicate that thismode particularly affects the single-ion anisotropy and the inter-planarsymmetric exchanges. By extracting the parameters of a magnetic modelHamiltonian from our first-principles results, we develop a phase diagram todescribe the magnetic structure as a function of the anisotropy and exchangeinteractions. This in turn allows us to explain the dependence of the magneticground state on the identity of the rare-earth ion and on theK\textsubscript{1} mode.

Influence of the triangular Mn-O breathing mode on the magnetic ordering in multiferroic hexagonal manganites