near a planar $\theta$-boundary {#charge_near_plane}
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Let us consider a point-like electric charge $q$ located at a distance $b>0$ from the $z=0$ plane, where we have chosen $a=0$. Also, the region $z<0$ is filled with a topologically non-trivial insulator whereas the region $%
z>0$ is the vacuum ($\theta_2=0$). For simplicity we choose the coordinates such that $x^{\prime }=y^{\prime }=0$. Therefore, the current density is $j^\mu(\mathbf{x}{}^{\prime })=q \eta ^{\mu} _{\; 0} \delta \left( x^{\prime
}\right) \delta \left( y^{\prime }\right) \delta \left( z^{\prime }-b\right) 
$. According to Eq. ([GreenMatrix]{}), the solution for this problem is $$A^{\mu }\left( \mathbf{x}\right) =qG_{\;0}^{\mu }\left( \mathbf{x},\mathbf{r}%
\right) ,  \label{SolPointCharge}$$where $\mathbf{r}=b\hat{\mathbf{e}}_{z}$. We first study the electrostatic potential. From Eq. (\[G00\]), $$\begin{aligned}
z>0\;\;\;\;\ &:&\;\;\;\;\ G_{\;0}^{0}\left( \mathbf{x},\mathbf{r}\right) =%
\frac{1}{|\mathbf{x}-\mathbf{r}|}-\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}%
^{2}}\frac{1}{|\mathbf{x}+\mathbf{r}|},  \label{G00z>} \\
z<0\;\;\;\;\ &:&\;\;\;\;\ G_{\;0}^{0}\left( \mathbf{x},\mathbf{r}\right) =%
\frac{4}{4+\tilde{\theta}^{2}}\frac{1}{|\mathbf{x} - \mathbf{r}|} .
\label{G00z<}\end{aligned}$$For $z>0$ the GF yields the electric potential $%
A^{0}\left( \mathbf{x}\right) =qG_{\;0}^{0}\left( \mathbf{x},\mathbf{r}%
\right)$ which can be interpreted as due to two point-like electric charges, one of strength $q$ at $\mathbf{%
r}$, and the other, the image charge, of strength $-q\tilde{\theta}^{2}/(4+%
\tilde{\theta}^{2})$, at the point $-\mathbf{r}$. For $z<0$ only one point-like electric charge appears, of strength $4q/(4+\tilde{\theta}^{2})$ located at $\mathbf{r}$.

&gt;From Eq. (\[SolPointCharge\]) we see that two components of 