| Summary: | We address the issue of the origin of the phonon thermal Hall effect in Tb_{3}Ga_{5}O_{12}, an intriguing property presumed to originate from magnetoelastic properties, and magnified in this compound by the non-Kramers nature of Tb^{3+} ions. Using neutron scattering, we have explored both the spin and lattice dynamics of Tb_{3}Ga_{5}O_{12}. Our experimental results show that the transition toward the magnetic ground state, below T_{N}=280 mK, is driven by the softening of an exciton, as expected in a two-singlet system like Tb_{3}Ga_{5}O_{12}. Low-energy excitations in the ordered phase are still excitons, whose dispersion throughout the Brillouin zone is driven by magnetic interactions. We have also discovered a mixing between specific phonon and exciton modes, this hybridization being evidenced through an intensity anomaly of the transverse acoustic phonons, as they cross low-energy crystal field excitations. Those experimental results can be comprehended by random phase approximation calculations, involving a Hamiltonian based on crystal electric field, dipolar interactions, and a coupling between phonons and the quadrupolar 4f electronic density.
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