Honeycomb lattice with multiorbital structure: Topological and quantum anomalous Hall insulators with large gaps

We construct a minimal four-band model for the two-dimensional topological insulators and quantum anomalous Hall insulators based on the $p_x$- and $p_y$-orbital bands in the honeycomb lattice. The multiorbital structure allows the atomic spin-orbit coupling which lifts the degeneracy between two sets of on-site Kramers doublets $j_z=\pm \frac{3}{2}$ and $j_z=\pm\frac{1}{2}$. Because of the orbital angular momentum structure of Bloch-wave states at $\Gamma$ and $K(K')$ points, topological gaps are equal to the atomic spin-orbit coupling strengths, which are much larger than those based on the mechanism of the $s-p$ band inversion.The energy spectra and eigen wave functions are solved analytically based on Clifford algebra. The competition among spin-orbit coupling $\lambda$, sublattice asymmetry $m$, and the N\'eel exchange field $n$ results in band crossings at $\Gamma$ and $K(K')$ points, which leads to various topological band structure transitions. The quantum anomalous Hall state is reached under the condition that three gap parameters $\lambda$, $m$, and $n$ satisfy the triangle inequality. Flat bands also naturally arise which allow a local construction of eigenstates. The above mechanism is related to several classes of solid state semiconductor.