Phonon-mediated spin-flipping mechanism in the spin ices Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$

To understand emergent magnetic monopole dynamics in the spin ices Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$, it is necessary to investigate the mechanisms by which spins flip in these materials. Presently there are thought to be two processes: quantum tunneling at low and intermediate temperatures and thermally activated at high temperatures. We identify possible couplings between crystal field and optical phonon excitations and construct a strictly constrained model of phonon-mediated spin flipping that quantitatively describes the high-temperature processes in both compounds, as measured by quasielastic neutron scattering. We support the model with direct experimental evidence of the coupling between crystal field states and optical phonons in Ho$_2$Ti$_2$O$_7$.