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Motivated by its role as a central pillar of current theories of dynamics ofspin ice in and out of equilibrium, we study the single-ion dynamics of themagnetic rare earth ions in their local environments, subject to the effectivefields set up by the magnetic moments they interact with. This effective fieldhas a transverse component with respect to the local easy-axis of the crystalelectric field, which can induce quantum tunnelling. We go beyond theprojective spin-1/2 picture and use instead the full crystal-field Hamiltonian.We find that the Kramers vs non-Kramers nature, as well as the symmetries ofthe crystal-field Hamiltonian, result in different perturbative behaviour atsmall fields ($\lesssim 1$ T), with transverse field effects being morepronounced in Ho$_{2}$Ti$_{2}$O$_{7}$ than in Dy$_{2}$Ti$_{2}$O$_{7}$.Remarkably, the energy splitting range we find is consistent with time scalesextracted from experiments. We also present a study of the static magneticresponse which highlights the anisotropy of the system in the form of anoff-diagonal $g$ tensor and we investigate the effects of thermal fluctuationsin the temperature regime of relevance to experiments. We show that there is anarrow yet accessible window of experimental parameters where the anisotropicresponse can be observed.

Single-ion anisotropy and magnetic field response in spin ice materials Ho$_{2}$Ti$_{2}$O$_{7}$ and Dy$_{2}$Ti$_{2}$O$_{7}$