Possibility of ferro-octupolar order in Ba$_2$CaOsO$_6$ assessed by X-ray magnetic dichroism measurements

Localized $5d^2$ electrons in a cubic crystal field possess multipoles such as electric quadrupoles and magnetic octupoles. We studied the cubic double perovskite Ba$_2$CaOsO$_6$ containing the Os$^{6+}$ ($5d^2$) ions, which exhibits a phase transition to a `hidden order' below $T^* \sim$ 50 K, by X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) at the Os $L_{2,3}$ edge. The cubic ligand-field splitting between the $t_{2g}$ and $e_g$ levels of Os $5d$ was deduced by XAS to be $\sim$4 eV. Ligand-field (LF) multiplet calculation under fictitious strong magnetic fields indicated that the exchange interaction between nearest-neighbor octupoles should be as strong as $\sim$1.5 meV if a ferro-octupolar order is stabilized in the `hidden-ordered' state, consistent with the exchange interaction of $\sim$1 meV previously predicted theoretically using model and density functional theory calculations. The temperature dependence of the XMCD spectra was consistent with a $\sim$18 meV residual cubic splitting of the lowest $J_{\rm eff} =$ 2 multiplet state into the non-Kramers $E_g$ doublet ground state and the $T_{2g}$ triplet excited state.