The honeycomb lattice with multi-orbital structure: topological and quantum anomalous Hall insulators with large gaps

We construct a minimal four-band model for the two-dimensional (2D) 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 \(K(K^\prime)\) 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. In the weak and intermediate regime of spin-orbit coupling strength, topological gaps are the global gap. 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 Néel exchange field \(n\) results in band crossings at \(\Gamma\) and \(K (K^\prime)\) 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 semiconducting materials.