High spin-Chern insulators with magnetic order.

As a topological insulator, the quantum Hall (QH) effect is indexed by the Chern and spin-Chern numbers C and Cspin. We have only Cspin = 0 or ± 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. It is shown that the spin-Chern number can takes an arbitrary high value for certain QH systems. This is a new type of topological insulators, which we may call high spin-Chern insulators. Samples may be provided by graphene on the SiC substrate with ferromagnetic order, transition-metal dichalcogenides with ferromagnetic order, transition-metal oxide with antiferromagnetic order and silicene with ferromagnetic order. Actually high spin-Chern insulators are ubiquitous in any systems with magnetic order. Nevertheless, the honeycomb system would provide us with unique materials for practical materialization.