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

We use a combination of symmetry analysis, phenomenological modeling, and first-principles density functional theory to explore the interplay between magnetic ground state detailed atomic structure in hexagonal rare-earth manganites. find that ordering is sensitive breathing mode distortion Mn O ions $ab$ plane, which described by ${\mathrm{K}}_{1}$ high-symmetry structure. Our calculations interactions indicate this particularly affects single-ion anisotropy interplanar symmetric exchanges. By extracting parameters model Hamiltonian from our results, we develop phase diagram describe as function exchange interactions. This turn allows us explain dependence on identity ion mode.