Ring-Exchange Interaction Effects on Magnons in Dirac Magnet CoTiO$_3$

The magnetic interactions that determine magnetic order and magnon energies typically involve only two spins. While rare, multi-spin interactions can also appear in quantum magnets and be the driving force in the ground state selection and in the nature of its excitations. By performing time-domain terahertz and magneto-Raman spectroscopy measurements combined with theoretical modeling, we determine the origin of the magnon excitation gap in Dirac antiferromagnet CoTiO$_3$. By adding a ring-exchange interaction in a hexagonal plaquette of the honeycomb lattice to both an XXZ spin model and to a low energy spin-orbital flavor wave model, a gap is generated in the magnon spectrum at the Brillouin zone center. With this addition, the flavor wave model reproduces a large swath of experimental results including terahertz, Raman, inelastic neutron scattering, and magnetization experiments.