Exchange couplings and edge states in two-dimensional topological insulators.

We consider two-dimensional honeycomb-lattice topological insulators of electrons which are exposed to exchange couplings at the edges, paying particular attention to the energy dispersions of edge states. The energy band structure of edge states is found to depend strongly on the direction of exchange couplings. Energy band splitting occurs for the exchange couplings perpendicular to the plane of topological insulators. In contrast, the parallel exchange couplings turn out to generate a finite energy gap in the edge-state bands for zigzag edges while armchair nanoribbons does not show such energy gaps. We also discuss the dependence of energy band structures of edge states on the magnetic structure of exchange couplings as well as the disorder effects on the edge-state density of states.