Quantization and $2π$ Periodicity of the Axion Action in Topological Insulators

Abstract: The Lagrangian describing the bulk electromagnetic response of a three-dimensional strong topological insulator contains a topological `axion' term of the form '\theta E dot B'. It is often stated (without proof) that the corresponding action is quantized on periodic space-time and therefore invariant under '\theta -> \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 wavefunctions in the underlying insulator.