Ab initio derivation of low-energy Hamiltonians for systems with strong spin-orbit interaction: Application to Ca 5 Ir 3 O 12

We present an ab initio derivation method for effective low-energy Hamiltonians of material with strong spin-orbit interactions. The effective Hamiltonian is described in terms of the Wannier function in the spinor form, and effective interactions are derived with the constrained random phase approximation (cRPA) method. Based on this formalism and the developed code, we derive an effective Hamiltonian of a strong spin-orbit interaction material ${\mathrm{Ca}}_{5}{\mathrm{Ir}}_{3}{\mathrm{O}}_{12}$. This system consists of three edge-shared ${\mathrm{IrO}}_{6}$ octahedral chains arranged along the $c$ axis, and the three Ir atoms in the $ab$ plane compose a triangular lattice. For such a complicated structure, we need to set up the Wannier spinor function under the local coordinate system. We found that a density-functional band structure near the Fermi level is formed by local ${d}_{xy}$ and ${d}_{yz}$ orbitals. Then, we constructed the ab initio ${d}_{xy}/{d}_{yz}$ model. The estimated nearest-neighbor transfer $t$ is close to 0.2 eV, and the cRPA on-site $U$ and neighboring $V$ electronic interactions are found to be 2.4--2.5 eV and 1 eV, respectively. The resulting characteristic correlation strength defined by $(U\ensuremath{-}V)/t$ is above 7, and thus this material is classified as a strongly correlated electron system. The on-site transfer integral involved in the spin-orbit interaction is 0.2 eV, which is comparable to the on-site exchange integrals $\ensuremath{\sim}0.2$ eV, indicating that the spin-orbit-interaction physics would compete with the Hund physics. Based on these calculated results, we discuss possible rich ground-state low-energy electronic structures of spin, charge, and orbitals with competing Hund, spin-orbit, and strong-correlation physics.