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Topological electronic structure of YbMg$_2$Bi$_2$ and CaMg$_2$Bi$_2$

Zintl compounds have been extensively studied for their outstanding thermoelectric properties, but their electronic structure remains largely unexplored. Here, we present a detailed investigation of the electronic structure of the isostructural thermopower materials YbMg$_2$Bi$_2$ and CaMg$_2$Bi$_2$ using angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT). The ARPES results show a significantly smaller Fermi surface and Fermi velocity in CaMg$_2$Bi$_2$ than in YbMg$_2$Bi$_2$. Our ARPES results also reveal that in the case of YbMg$_2$Bi$_2$, Yb-4$f$ states reside well below the Fermi level and likely have a negligible impact on transport properties. To properly model the position of 4$f$-states, as well as the overall electronic structure, a Hubbard $U$ at the Yb sites and spin-orbit coupling (SOC) have to be included in the DFT calculations. Interestingly, the theoretical results reveal that both materials belong to a $Z_2$ topological class and host robust topological surface states around $E_\mathrm {F}$. Due to the intrinsic hole doping, the topological states reside above the Fermi level, inaccessible by ARPES. Our results also suggest that in addition to SOC, vacancies and the resulting hole doping play an important role in the transport properties of these materials.