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Two-dimensional topological semimetal states in monolayers Cu$_2$Ge, Fe$_2$Ge, and Fe$_2$Sn

Recent experimental realizations of the topological semimetal states in several monolayer systems are very attractive because of their exotic quantum phenomena and technological applications. Based on first-principles density-functional theory calculations including spin-orbit coupling, we here explore the drastically different two-dimensional (2D) topological semimetal states in three monolayers Cu$_2$Ge, Fe$_2$Ge, and Fe$_2$Sn, which are isostructural with a combination of the honeycomb Cu or Fe lattice and the triangular Ge or Sn lattice. We find that (i) the nonmagnetic (NM) Cu$_{2}$Ge monolayer having a planar geometry exhibits the massive Dirac nodal lines, (ii) the ferromagentic (FM) Fe$_2$Ge monolayer having a buckled geometry exhibits the massive Weyl points, and (iii) the FM Fe$_2$Sn monolayer having a planar geometry and an out-of-plane magnetic easy axis exhibits the massless Weyl nodal lines. It is therefore revealed that mirror symmetry cannot protect the four-fold degenerate Dirac nodal lines in the NM Cu$_{2}$Ge monolayer, but preserves the doubly degenerate Weyl nodal lines in the FM Fe$_{2}$Sn monolayer. Our findings demonstrate that the interplay of crystal symmetry, magnetic easy axis, and band topology is of importance for tailoring various 2D topological states in Cu$_2$Ge, Fe$_2$Ge, and Fe$_2$Sn monlayers.