Flat optical conductivity in the topological kagome magnet TbMn$_6$Sn$_6$

Kagome magnet TbMn$_6$Sn$_6$ is a new type of topological material that is known to support exotic quantum magnetic states. Experimental work has identified that TbMn$_6$Sn$_6$ hosts Dirac electronic states that could lead to topological and Chern quantum phases, but the optical response of the Dirac fermions of TbMn$_6$Sn$_6$ and its properties remain to be explored. Here, we perform optical spectroscopy measurement combined with first-principles calculations on single-crystal sample of TbMn$_6$Sn$_6$ to investigate the associated exotic phenomena. TbMn$_6$Sn$_6$ exhibits frequency-independent optical conductivity spectra in a broad range from 1800 to 3000 cm$^{-1}$ (220-370 meV) in experiments. The theoretical band structures and optical conductivity spectra are calculated with several shifted Fermi energy to compare with the experiment. The theoretical spectra with 0.56 eV shift for Fermi energy are well consistent with our experimental results. Besides, the massive quasi-two-dimensional (quasi-2D) Dirac bands, which have linear band dispersion in $k_x$-$k_y$ plane and no band dispersion along the $k_z$ direction, exist close to the shifted Fermi energy. According to tight-binding model analysis, the quasi-2D Dirac bands give rise to a flat optical conductivity, while its value is smaller than, about one tenth of, that from the calculations and experiments. It indicates that the other trivial bands also contribute to the flat optical conductivity.