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We report an infrared spectroscopy study of the axion topological insulatorcandidate EuIn$_2$As$_2$ for which the Eu moments exhibit an A-typeantiferromagnetic (AFM) order below $T_N \simeq 18 \mathrm{K}$. The low energyresponse is composed of a weak Drude peak at the origin, a pronouncedinfrared-active phonon mode at 185 cm$^{-1}$ and a free carrier plasma edgearound 600 cm$^{-1}$. The interband transitions start above 800 cm$^{-1}$ andgive rise to a series of weak absorption bands at 5\,000 and 12\,000 cm$^{-1}$and strong ones at 20\,000, 27\,500 and 32\,000 cm$^{-1}$. The AFM transitiongives rise to pronounced anomalies of the charge response in terms of acusp-like maximum of the free carrier scattering rate around $T_N$ and largemagnetic splittings of the interband transitions at 5\,000 and 12\,000cm$^{-1}$. The phonon mode at 185 cm$^{-1}$ has also an anomalous temperaturedependence around $T_N$ which suggests that it couples to the fluctuations ofthe Eu spins. The combined data provide evidence for a strong interactionamongst the charge, spin and lattice degrees of freedom.

Infrared study of the interplay of charge, spin, and lattice excitations in the magnetic topological insulator EuIn$_2$As$_{2}$