  With infrared spectroscopy we studied the bulk electronic properties of the
topological antiferromagnet MnBi$_2$Te$_4$ with $T_N \simeq 25~\mathrm{K}$.
With the support of band structure calculations, we assign the intra- and
interband excitations and determine the band gap of $E_g \approx$ 0.17 eV. We
also obtain evidence for two types of conduction bands with light and very
heavy carriers. The multiband free carrier response gives rise to an unusually
strong increase of the combined plasma frequency, $\omega_{\mathrm{pl}}$, below
300 K. The band reconstruction below $T_N$, yields an additional increase of
$\omega_{\mathrm{pl}}$ and a splitting of the transition between the two
conduction bands by about 54 meV. Our study thus reveals a complex and strongly
temperature dependent multi-band low-energy response that has important
implications for the study of the surface states and device applications.
