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Motivated by studies of coexisting electron correlation and spin-orbitcoupling effect in Na$_2$IrO$_3$ and a recent experiment of its 4d analogueLi$_2$RhO$_3$, we performed first-principles calculations of the rhodium oxidecompound. The experimentally observed ground state of Li$_2$RhO$_3$ can berecovered only if both spin-orbit coupling and on-site Coulomb interaction aretaken into consideration. Within the proper $U$ range for 4d-orbitals($2\leqslant U\leqslant 4$ eV), the ground state of Li$_2$RhO$_3$ could beeither zigzag-AFM or stripy-AFM, both yielding energy gap close to experimentalobservation. Furthermore, the total energy differences between the competingmagnetic phases are $\leqslant 3$ meV/Rh within $2\leqslant U\leqslant 4$ eV,manifesting strong magnetic frustration in the compound. Finally, the phaseenergy of Li$_2$RhO$_3$ cannot be fitted with the two-dimensionalHeisenberg-Kitaev model involving only the nearest neighbor interactions, andwe propose that inter-layer interactions may be responsible for thediscrepancy.

Electronic structure of relativistic Mott insulator Li$_2$RhO$_3$