Title: Origin of Charge Density Wave in Topological Semimetals SrAl4 and EuAl4

Abstract: Topological semimetals in BaAl4-type structure show many interesting behaviors, such as charge density wave (CDW) in SrAl4 and EuAl4, but not the isostructural and isovalent BaAl4, SrGa4 and BaGa4. Here using Wannier functions based on density functional theory, we calculate the susceptibility functions with millions of k-points to reach the small q-vector and study the origin and driving force behind the CDW. Our comparative study reveals that the origin of the CDW in SrAl4 and EuAl4 is the strong electron-phonon coupling interaction for the transverse acoustic mode at small q-vector along the \{Gamma}-Z direction besides the maximum of the real part of the susceptibility function from the nested Fermi surfaces of the Dirac-like bands, which explains well the absence of CDW in the other closely related compounds in a good agreement with experiment. We also connect the different CDW behaviors in the Al compounds to the macroscopic elastic properties.