Infrared study of the interplay of charge, spin, and lattice excitations in the magnetic topological insulator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Eu</mml:mi><mml:msub><mml:mi>In</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>As</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>

We report an infrared spectroscopy study of the axion topological insulator candidate EuIn$_2$As$_2$ for which Eu moments exhibit A-type antiferromagnetic (AFM) order below $T_N \simeq 18 \mathrm{K}$. The low energy response is composed a weak Drude peak at origin, pronounced infrared-active phonon mode 185 cm$^{-1}$ and free carrier plasma edge around 600 cm$^{-1}$. interband transitions start above 800 give rise to series absorption bands 5\,000 12\,000 strong ones 20\,000, 27\,500 32\,000 AFM transition gives anomalies charge in terms cusp-like maximum scattering rate $T_N$ large magnetic splittings has also anomalous temperature dependence suggests that it couples fluctuations spins. combined data provide evidence interaction amongst charge, spin lattice degrees freedom.