Quantization and $2\pi$ periodicity of the axion action in topological insulators

The Lagrangian describing the bulk electromagnetic response of a three-dimensional strong topological insulator contains a topological “axion” term of the form $\theta \mathbf{E}\cdot \mathbf{B}$. It is often stated (without proof) that the corresponding action is quantized on periodic space time and therefore invariant under $\theta \to \theta +2\pi$. Here we provide a simple, physically motivated proof of the axion action quantization on the periodic space time, assuming only that the vector potential is consistent with single valuedness of the electron wave functions in the underlying insulator.

Figure 1

(Color online) (a) A magnetic field increasing linearly with time and confined to a torus hole produces a uniform and constant electric field along the torus. (b) A closed path on the torus can be thought of as enclosing two areas, ${\Omega }_1$ and ${\Omega }_2$.