Table 2.

Space groups (SGs) that admit Dirac points with HOFA states derived from the QI introduced in ref. ²³.

Space groups admitting dirac points with HOFA states
Point group name	Point group symbol	SG numbers
C4v	$4mm{1}^{\prime }$	99–110
D4h	$4∕mmm{1}^{\prime }$	123–142
Oh	$m\overline{3}m{1}^{\prime }$	221–230

All of these SGs have point groups that contain C4v. We obtain this list of SGs by combining the nested Jackiw–Rebbi formulation of the QI in Supplementary Note 6 with an analysis of the crystallographic rod groups in Supplementary Note 12. In all of these SGs, semimetals with Dirac points along lines with 4 mm or $4∕m^{\prime }mm$ symmetry will exhibit intrinsic HOFA states when cut into nanorods that preserve fourfold axes and are thick compared with the in-plane lattice spacing. This list is a complete enumeration of the SGs that permit Dirac semimetals with HOFA states directly related to the QI introduced in ref. ²³; alternative realizations of Dirac and Weyl semimetals with HOFA states derived from other 2D magnetic insulators with corner states are also possible (refs. ^32,37 and Supplementary Note 12).