Figure 2: 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 (${\sigma }_{xy}$) 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 $\text{arctan}(1/\alpha )$ and $\text{arctan}(\alpha )$, respectively. These angles are independent of material details. $E_o$, $E_R$, and $E_T$ represent incident, reflected, and transmitted beams. ${\theta }_F$ and ${\theta }_K$ are the Faraday and Kerr angles, respectively.