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We report our theoretical predictions on the linear magnetoelectric (ME)effects originating from odd-parity multipoles associated with spontaneous spinand orbital ordering on a diamond structure. We derive a two-orbital model for$d$ electrons in $e_g$ orbitals by including the effective spin-orbit couplingwhich arises from the mixing between $e_g$ and $t_{2g}$ orbitals. We show thatthe model acquires a net antisymmetric spin-orbit coupling once staggered spinand orbital orders occur spontaneously. The staggered orders are accompanied byodd-parity multipoles: magnetic monopole, quadrupoles, and toroidal dipoles. Weclassify the types of the odd-parity multipoles according to the symmetry ofthe spin and orbital orders. Furthermore, by computing the ME tensor using thelinear response theory, we show that the staggered orders induce a variety ofthe linear ME responses. We elaborate all possible ME responses for eachstaggered order, which are useful to identify the order parameter and to detectthe odd-parity multipoles by measuring the ME effects. We also elucidate theeffect of lowering symmetry by a tetragonal distortion, which leads to richerME responses. The implications of our results are discussed for $5d$ transitionmetal oxides, $A$OsO$_4$ ($A=$ K, Rb, and Cs), in which the order parametersare not fully identified.

physical review. b./physical review. b

Emergent odd-parity multipoles and magnetoelectric effects on a diamond  structure: implication to 5$d$ transition metal oxides $A$OsO$_4$ ($A=$ K,  Rb, and Cs)