  In this study, ultrafast optical spectroscopy was employed to elucidate the
intricate topological features of EuIn$_2$As$_2$, a promising candidate for a
magnetic topological-crystalline axion insulator. Our investigation, focusing
on the real-time evolution of topological states, unveiled a narrow surface
magnetic gap (2$\Delta_0$ $\simeq$ 8.2 meV)) emerging at the antiferromagnetic
transition temperature ($T_N$ $\approx$ 16 K). Below $T_N$, two extremely
low-energy collective modes, $\omega_1$ and $\omega_2$, with frequencies of
$\sim$9.9 and 21.6 GHz at $T$ = 4 K, respectively, were observed, exhibiting
strong temperature dependence. $\omega_1$ correlates with an acoustic phonon,
while $\omega_2$ is associated with a magnon. The results suggest that
EuIn$_2$As$_2$ has the potential to manifest a magnetic topological-crystalline
axion insulator, presenting a small magnetic energy gap on the (001) surface.
The findings further our understanding of the interplay between magnetism and
topology in this material, showcasing its potential for applications in quantum
information processing and spintronics.
