Spin phonon induced magnetic soft mode in triangular antiferromagnet h-RMnO$_3$

The relief of geometrical magnetic frustration by spin-lattice coupling is an extensively studied subject: For example, theory shows that in triangular lattice antiferromagnets, the spin-phonon coupling can stabilize a collinear magnet order over the non-collinear 120$^{\circ}$ order. We report inelastic neutron scattering measurements on the triangular lattice antiferromagnets (Y/Lu)MnO$_3$, showing evidence of magnon-phonon hybridization at the Brillouin zone boundary. Furthermore, a magnetic soft mode is observed at lower energy at the same momentum transfer. The exchange striction model within the linear approximation qualitatively explains the observed features while the 1/$S$ expansion calculation shows that a third of the observed softening is caused by magnon-magnon interactions. Our results demonstrate how the spin-phonon coupling $and$ quantum effect cooperatively develops the roton-like minimum in triangular lattice antiferromagnets with a 120$^{\circ}$ structure.