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As a topological insulator, the quantum Hall (QH) effect is indexed by theChern and spin-Chern numbers $\mathcal{C}$ and $\mathcal{C}_{\text{spin}}$. Wehave only $\mathcal{C}_{\text{spin}}=0$ or $\pm \frac{1}{2}$ in conventional QHsystems. We investigate QH effects in generic monolayer honeycomb systems. Wesearch for spin-resolved characteristic patterns by exploring Hofstadter'sbutterfly diagrams in the lattice theory and fan diagrams in the low-energyDirac theory. The Chern and spin-Chern numbers are calculated based on thebulk-edge correspondence in the lattice theory and on the Kubo formula in theDirac theory. It is shown that the spin-Chern number can takes an arbitraryhigh value for certain QH systems in coexistence with buckled structure andmagnetic order. This is a new type of topological insulators. Samples may beprovided by silicene with ferromagnetic order and transition-metal oxide withantiferromagnetic order.

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