Quantum anomalous Hall effect in d -electron kagome systems: Chern insulating states from transverse spin-orbit coupling

The possibility of the quantum anomalous Hall effect (QAHE) in two-dimensional kagome systems with $d$-orbital electrons is studied within a multiorbital tight-binding model. We concentrate on the case of isotropic Slater-Koster integrals which is realized in a recently discovered class of metal-organic frameworks ${TM}₃{\mathrm{C}}₆{\mathrm{O}}₆$ with transition metals ($\mathit{TM}$) in the beginning of the $3d$ series. Furthermore, in the absence of exchange-type spin-orbit coupling, only isotropic Slater-Koster integrals give a perfect flatband in addition to the two dispersive bands hosting relativistic (Dirac) and quadratic band crossing points at high symmetry spots in the Brillouin zone. A quantized topological invariant requires a flux-creating spin-orbit coupling, giving Chern number (per spin sector) $C=1$ not only from the familiar Dirac points at the six corners of the Brillouin zone, but also from the quadratic band crossing point at the center $\mathrm{\ensuremath{\Gamma}}$. In the case of isotropic Slater-Koster integrals the on-site spin-orbit coupling (SOC) is ineffective in creating the QAHE and it is only the transfer or exchange-type SOC which can lead to a QAHE. Surprisingly, this QAHE comes from the nontrivial effective flux induced by the transverse part of the spin-orbit coupling, exhibited by electrons in the $d$-orbital state with ${m}_l=0$ (${d}_{{z}²}$ orbital), in stark contrast to the more familiar form of QAHE due to the $d$ orbitals with ${m}_l\ensuremath{\ne}0$, driven by the Ising part of spin-orbit coupling. The $C=1$ Chern plateau (per spin sector) due to the Dirac point extends over a smaller region of Fermi energy than that due to quadratic band crossing. Our result hints at the promising potential of kagome $d$-electron systems as a platform for dissipationless electronics by virtue of its unique QAHE.