Van Hove Singularity Arising from Mexican-Hat-Shaped Inverted Bands in the Topological Insulator Sn-doped Bi$_{1.1}$Sb$_{0.9}$Te$_{2}$S

The optical properties of Sn-doped Bi$_{1.1}$Sb$_{0.9}$Te$_{2}$S, the most bulk-insulating topological insulator thus far, have been examined at different temperatures over a broad frequency range. No Drude response is detected in the low-frequency range down to 30~cm$^{-1}$, corroborating the excellent bulk-insulating property of this material. Intriguingly, we observe a sharp peak at about 2,200~cm$^{-1}$ in the optical conductivity at 5~K. Further quantitative analyses of the line shape and temperature dependence of this sharp peak, in combination with first-principles calculations, suggest that it corresponds to a van Hove singularity arising from Mexican-hat-shaped inverted bands. Such a van Hove singularity is a pivotal ingredient of various strongly correlated phases.