First-principles study of the magnetic interactions in honeycomb Na2IrO3

Honeycomb iridate $\mathrm{N}{\mathrm{a}}₂\mathrm{Ir}{\mathrm{O}}₃$, a ${J}_{\mathrm{eff}}=1/2$ magnet, is a potential platform for realizing quantum spin liquid. Many experiments have shown that its magnetic ground state is a zigzag antiferromagnetic order. However, there is still a lack of consensus on the theoretical model explaining such order, since its second-nearest-neighbor (NN) and long-range third-NN magnetic interactions are highly unclear. By properly considering the orbital moments achieved through constraining their directions in first-principles calculations, we obtain that the relative angle between orbital and spin moments is fairly small and in the order of several degrees, which thus validates the ${J}_{\mathrm{eff}}=1/2$ state in $\mathrm{N}{\mathrm{a}}₂\mathrm{Ir}{\mathrm{O}}₃$. Surprisingly, we find that the long-range third-NN Heisenberg interactions are sizable, whereas the second-NN magnetic interactions are negligible. Furthermore, we show that sizable long-range third-NN Heisenberg interactions closely correlate with the appreciable distribution of Wannier orbitals of ${J}_{\mathrm{eff}}=1/2$ states over the three NN Ir atoms. Based on our study, we propose a minimal ${J}₁\text{\ensuremath{-}}{K}₁\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}₁\text{\ensuremath{-}}{J}₃$ model in which the magnetic excitations have an intensity peak at 5.6 meV, consistent with the inelastic neutron-scattering experiment [Phys. Rev. Lett. 108, 127204 (2012)]. The present work demonstrates again that constraining orbital moments in first-principles calculations is a powerful way to investigate the intriguing magnetism in ${J}_{\mathrm{eff}}=1/2$ magnets, and it paves the way toward gaining a deep insight into the novel magnetism discovered in the honeycomb ${J}_{\mathrm{eff}}=1/2$ magnets.