Effective field theory description of topological crystalline insulators

We propose a phenomenological theory for topological crystalline insulators with time reversal and $C_4$ symmetries. First, we introduce a fictitious space and transformation of electromagnetic field operators. This transformation leaves the speed of light unchanged but changes the elementary charge to $\sqrt{2}e$. Then we formulate the theory of topological crystalline insulators in terms of transformed fields in this fictitious space as 3D BF theory containing $\pi$-flux excitations. It is known that a 3D BF theory with half flux quantum excitations describes low energy properties of time reversal invariant insulators. By making an inverse transform we recover the effective field theory in original space. It turns out that this field theory contains quarter flux quantum excitations.