Magneto-optical effect in a 3D topological insulator (TI). (Top) Applying a weak magnetic field to the surfaces of a 3D topological insulator (blue slab) causes an energy gap to open in the spectrum of electronic surface states (grey). This amounts to a half-integer Hall conductivity (σxy) on each of its surfaces. (Bottom) The rotation of light that is reflected from (Kerr effect) or transmitted through (Faraday effect) a 3D topological insulator in a magnetic field is given by arctan(1/α) and arctan(α), respectively. These angles are independent of material details. Eo, ER, and ET represent incident, reflected, and transmitted beams. θF and θK are the Faraday and Kerr angles, respectively.