Spinless Hourglass Nodal-line Semimetals

Nodal-line semimetal have degeneracy along nodal lines where the band gap is closed. In many cases, the nodal lines appear accidentally, and it is impossible to determine whether the nodal lines appear or not, only from the symmetry and the electron filling. For spinless systems, we show that in specific space groups at 8N+ 4 fillings, presence of the nodal lines is required. In this case the shape of the band structure around these nodal lines is like an hourglass, and we call this a spinless hourglass nodal-line semimetal. We establish a list of all the centrosymmetric space groups, under which nonmagnetic spinless systems have hourglass nodal lines, and illustrate where the nodal lines are located. We propose that Al₃FeSi₂ is one of the hourglass nodal-line semimetals. Because of inversion symmetry and time-reversal symmetry, these nodal lines are also characterized by π Berry phase (or Zak phase). If inversion symmetry is broken, this Zak phase might yield electric polarization. We will discuss polarization after the hourglass nodal lines are gapped out by inversion symmetry breaking. We also discuss some specific materials where the hourglass nodal lines are gapped out by spontaneous breaking of inversion symmetry.
[1]R. Takahashi, M. Hirayama and S. Murakami, PRB 96, 155206