of classical $\theta$-ED [@ZH] and in the quantum vacuum framework [@Canfora:2011fd]. For related analyses, in several contexts, see [@related]. The latter approach has been used to describe TIs. Concretely, the low-energy limit of the electrodynamics of TIs can be described by extending Maxwell electrodynamics by Eq. (\[FF\*\]), originally formulated in 4+1 dimensions but appropriately adapted to lower dimensions by dimensional reduction [@Qi:2008ew]. Thus, $\theta$-ED as a topological field theory (TFT), serves as model for many theoretical [@theo_topoins] and experimental realizations for studying detailed properties of topological states of quantum matter [@tech-apps; @topo_reviews].

The formulation of $\theta $-ED pursued in this work can be considered as a particularly simple version of the so called Janus field theories [@CFKS; @DEG; @Chen; @Witten2; @Kim1; @Kim2]. Generally speaking, such theories are characterized by having space-time dependent coupling constants, as is $\theta$ in our model. They have been actively explored in the context of the AdS/CFT correspondence. Nevertheless, as we have already mentioned, in the case of $\theta $-ED this idea is applied to a simpler but more realistic system, that constitutes an effective low energy theory which allows to compute the response of topological insulators to arbitrary external sources and currents in a planar 