Magnetic ordering on edges of topological insulator

Based on first-principles calculations, we show that the Bi$_{\mathrm{2}}$Se$_{\mathrm{3}}$ crystal, which is a prototypical topological insulator, with different crystal-face orientation surfaces generates a built-in electric field around the facet edges due to the work function differences. For a given broken time reversal symmetry in the crystal, the electric field, in turn, induces a net magnetic ordering along the edges by the topological magnetoelectric coupling. The predicted magnetic ordering depending on the work function differences between facets would be a unique manifestation of the axion electrodynamics in real solids and suggests a route to reveal novel properties of macroscopic topological edge states.