astonishing precision with which the Hall conductance of a sample is quantized [@von; @Klitzing:1980kg], despite the varying irregularities and geometry of the sample, turned out to have a topological origin. The reason for this lies in the band structure of the sample, but ultimately, the Hall conductance can be expressed as an invariant integral over the frequency in momentum space, more precisely as an integral of the Berry curvature over the Brillouin zone [@SQ; @Shen], inasmuch as the genus of a manifold can be expressed in terms of an invariant integral of the local curvature over the surface enclosing it. This quantity plays the role of a topological order parameter uniquely determining the nature of the quantum state, as the order parameter in Landau-Ginzburg effective field theory determines the usual phases of quantum matter.

In this work we will be concerned with a simple case of CS theories, to which we will refer as $\theta$-electrodynamics or simply $\theta$-ED and it amounts to extending Maxwell electrodynamics by a parity violating term of the form $$\Delta \mathcal{L}_\theta = \theta (\alpha / 4 \pi^2) \mathbf{E} \cdot 
\mathbf{B} = - {\frac{\theta }{4}} (\alpha / 4 \pi^2) F_{\mu \nu} \tilde
F^{\mu \nu},\label{FF*}$$ where $\tilde{F}^{\mu \nu 