Chiral Surface Modes in Three-Dimensional Topological Insulators

Where chiral modes should appear is an essential question for the quantum anomalous Hall (QAH) effect in three-dimensional topological insulators (3DTIs). In this Letter, we show that in a slab of ferromagnetic 3DTI subjected to a uniform exchange field normal to its top and bottom surfaces, the QAH effect creates a single chiral surface mode delocalized on the side faces. In a nonmagnetic 3DTI, analogously, delocalized helical modes consisting of a pair of oppositely propagating chiral surface modes are produced by the quantum spin Hall effect.