Fig. 1: Schematic demonstration for the wallpaper fermions and third-order topology.

a Schematic of wallpaper group p4g with two perpendicular glide lines \({\mathbb G}_{x/y}\) indicated with red/blue dashed lines. b Two categories of band crossings induced by \({\Bbb G}_{x/y}\) along the invariant paths, which are constrained by the commutation relations of \({\Bbb G}_{x/y}\) and \({\Bbb T}{\Bbb G}_{y/x}\). c When \({\Bbb G}_{x/y}\) anticommuting with \({\Bbb T}{\Bbb G}_{y/x}\), a fourfold-degenerate Dirac point appears at the zone boundary k = π, i.e., at M as in b. d When \({\Bbb G}_{x/y}\) commute with \({\Bbb T}{\Bbb G}_{y/x}\), the band crossing is twofold degenerate and an hourglass-shaped dispersion with an internal partner switching for each quadruplet appears. e Due to the \({\Bbb G}_{x/y}\) eigenvalues \(g_{ \pm x/y} = \pm ie^{ - ik_{y/x}/2}\), surface states cannot go to the origin after a period of 2π, but a periodicity of 4π, featuring a Möbius fermion character. f Schematic of a helical third-order topological insulator with time-reversal polarized octupole moments.