Magnitude of quantum effects in classical spin ices

The pyrochlore spin ice compounds ${\mathrm{Dy}}_{\text{2}}{\mathrm{Ti}}_{\text{2}}{\mathrm{O}}_{\text{7}}$ and ${\mathrm{Ho}}_{\text{2}}{\mathrm{Ti}}_{\text{2}}{\mathrm{O}}_{\text{7}}$ are well described by classical Ising models down to low temperatures. Given the empirical success of this description, the question of the importance of quantum effects in these materials has been mostly ignored. We show that the common wisdom that the strictly Ising moments of isolated ${\mathrm{Dy}}^{\text{3+}}$ and ${\mathrm{Ho}}^{\text{3+}}$ ions imply Ising interactions is too naïve; a more complex argument is needed to explain the close agreement between theory and experiment. From a microscopic picture of the interactions in rare-earth oxides, we show that the high-rank multipolar interactions needed to induce quantum effects in these two materials are generated only very weakly by superexchange. Using this framework, we formulate an estimate of the scale of quantum effects in ${\mathrm{Ho}}_{\text{2}}{\mathrm{Ti}}_{\text{2}}{\mathrm{O}}_{\text{7}}$ and ${\mathrm{Dy}}_{\text{2}}{\mathrm{Ti}}_{\text{2}}{\mathrm{O}}_{\text{7}}$, finding it to be well below experimentally relevant temperatures. We discuss the implications of these results for realizing quantum spin ice in other materials.

Figure 1

Process 1 involves intermediate states with only a single hole on the oxygen site. Scales roughly as $\sim t^4{(U_f+{\mathrm{\Delta }}_{pf})}^{-2}{\left(2U_f\right)}^{-1}$.

Figure 2

Process 2 involves an intermediate state with two holes on the oxygen site. Scales roughly as $\sim t^4{(U_f+{\mathrm{\Delta }}_{pf})}^{-2}{(2U_f+2{\mathrm{\Delta }}_{pf}+U_p)}^{-1}$.

Figure 3

Process 3 involves an intermediate state with two holes on the oxygen site. Scales roughly as $\sim t^4{(U_f+{\mathrm{\Delta }}_{pf})}^{-2}{(2U_f+2{\mathrm{\Delta }}_{pf}+U_p)}^{-1}$.