Motivated by its role as a central pillar of current theories of the dynamics of spin ice in and out of equilibrium, we study the single-ion dynamics of the magnetic rare-earth ions in their local environments, subject to the effective fields set up by the magnetic moments with which they interact. This effective field has a transverse component with respect to the local easy axis of the crystal electric field, which can induce quantum tunneling. We go beyond the projective spin-1/2 picture and use instead the full crystal-field Hamiltonian. We find that the Kramers versus non-Kramers nature, as well as the symmetries of the crystal-field Hamiltonian, result in different perturbative behavior at small fields ($\lesssim 1\mathrm{T}$), with transverse field effects being more pronounced in ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ than in ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Remarkably, the energy splitting range we find is consistent with time scales extracted from experiments. We also present a study of the static magnetic response, which highlights the anisotropy of the system in the form of an off-diagonal $g$ tensor, and we investigate the effects of thermal fluctuations in the temperature regime of relevance to experiments. We show that there is a narrow but accessible window of experimental parameters where the anisotropic response can be observed.

• Received 15 June 2015

DOI:https://doi.org/10.1103/PhysRevB.92.155120