Fig. 3: Magneto-dielectric effects at low temperature in Tb₂Ti₂O₇ and Ho₂Ti₂O₇ compared to Khomskii’s model.

Magnetic field variation of the dielectric permittivity in Tb₂Ti₂O₇ (a) and Ho₂Ti₂O₇ (b) below 30 K probed through the magnetic field variation of the permittivity with respect to its zero field value at 3 K and 2.5 K respectively. The electric field E is applied parallel to the static magnetic field H along the three main directions of the cubic pyrochlore structure. Note the different scales of \(\Delta {\varepsilon }_{{\rm{r}}}^{{\prime} }\) for the two compounds. For Ho₂Ti₂O₇, the applied magnetic field was corrected from the demagnetization field to allow a correct comparison with the spin ice phase diagram. c Monte Carlo calculations of the dielectric susceptibility (ε_r − 1) using Khomskii’s model at the same temperatures and for the same field orientations as the measurements. It is normalized at 1 at 2.5 K and zero field. The vertical dashed lines in (c) point out to the changes in regime of the calculated dielectric susceptibility at 2.5 K. These lines are reported on the Ho₂Ti₂O₇ measurements in panel (b) and on the magnetization measurements in Fig. 1c.