Two types of fermionic triple-point quasiparticles. (a) and (c) Type-A triple-points are connected by a single nodal line, where conduction and valence bands are degenerate (shown in black). (b) and (d) Type-B triple-points are accompanied by four such nodal lines, shown in black, green, blue, and red. The latter three occur in the mirror-symmetric planes in momentum space. The grey circles in (c) and (d) indicate paths for the Berry phase calculation. (e) and (f) Band structures around type-A and type-B triple-points along the C₃ axis, respectively. Here Λ 6 represents the double degenerate band (double representation of C 3 v ), while Λ 4 , 5 correspond to two one-dimensional representations. The black lines in (e) and (f) mark the region of the band structure that produces the nodal lines shown in black in panels (c) and (d). (g) and (h) Band structures around type-A and type-B triple-points in a mirror symmetric plane orthogonal to k_z, respectively. The dashed green lines in (e) and (f) mark the momentum k_z used in panels (g) and (h). Red (blue) color in panels (e)–(h) corresponds to occupied (unoccupied) bands assuming that the Fermi level is exactly between the pair of triple-points. Figure partially reused from Ref. [21].