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

Motivated by studies of coexisting electron correlation and spin-orbit coupling effect in Na$_2$IrO$_3$ and a recent experiment of its 4d analogue Li$_2$RhO$_3$, we performed first-principles calculations of the rhodium oxide compound. The experimentally observed ground state of Li$_2$RhO$_3$ can be recovered only if both spin-orbit coupling and on-site Coulomb interaction are taken 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 be either zigzag-AFM or stripy-AFM, both yielding energy gap close to experimental observation. Furthermore, the total energy differences between the competing magnetic phases are $\leqslant 3$ meV/Rh within $2\leqslant U\leqslant 4$ eV, manifesting strong magnetic frustration in the compound. Finally, the phase energy of Li$_2$RhO$_3$ cannot be fitted with the two-dimensional Heisenberg-Kitaev model involving only the nearest neighbor interactions, and we propose that inter-layer interactions may be responsible for the discrepancy.