Quantum Hall Effects with High Spin-Chern Numbers in Buckled Honeycomb Structure with Magnetic Order

As a topological insulator, the quantum Hall (QH) effect is indexed by the Chern and spin-Chern numbers $\mathcal{C}$ and $\mathcal{C}_{\text{spin}}$. We have only $\mathcal{C}_{\text{spin}}=0$ or $\pm \frac{1}{2}$ in conventional QH systems. We investigate QH effects in generic monolayer honeycomb systems. We search for spin-resolved characteristic patterns by exploring Hofstadter's butterfly diagrams in the lattice theory and fan diagrams in the low-energy Dirac theory. The Chern and spin-Chern numbers are calculated based on the bulk-edge correspondence in the lattice theory and on the Kubo formula in the Dirac theory. It is shown that the spin-Chern number can takes an arbitrary high value for certain QH systems in coexistence with buckled structure and magnetic order. This is a new type of topological insulators. Samples may be provided by silicene with ferromagnetic order and transition-metal oxide with antiferromagnetic order.