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Topological nodal-line semimetals are exotic conductors that hostsymmetry-protected conducting nodal-lines in their bulk electronic spectrum andnontrivial drumhead states on the surface. Based on first-principlescalculations and an effective model analysis, we identify the presence oftopological nodal-line semimetal states in the TT'X family of compounds (T, T'= transition metal, X= Si, or Ge) in the absence of spin-orbit coupling (SOC).Taking ZrPtGe as an exemplar system, we show that this material harbors asingle nodal line on the $k_y=0$ plane, which is protected by the $M_y$ mirrorplane symmetry. Surface electronic structure calculations further reveal theexistence of a drumhead surface state nested inside the nodal line projectionon the (010) surface with a saddle-like energy dispersion. When the SOC isincluded, the nodal line gaps out and the system transitions to a strongtopological insulator state with $Z_2=(1;000)$. The topological surface stateevolves from the drumhead surface state via the sharing of its saddle-likeenergy dispersion within the bulk energy gap. These features differ remarkablyfrom those of the currently known topological surface states in topologicalinsulators such as Bi$_2$Se$_3$ with Dirac-cone-like energy dispersions.

Saddle-like topological surface states on the TT'X family of compounds (T, T' = Transition metal, X= Si, Ge)