Extended Data Fig. 7: Low-energy spectra of the tight binding model in absence of first-order topology.

Here we delineate the step edge dispersion in a tight-binding system that differs from α-As in that it only has a double band inversion at the Γ point of the bulk Brillouin zone, and not an extra single band inversion at each of the three L, L′, L″ points as is the case for α-As. Correspondingly, the model only has nontrivial higher-order topology, but no first-order topology. As derived in our Supplementary Information, such a situation implies a gapped step edge dispersion, however, the gapped modes are precursors of hinge modes localized near the step edges. This situation is schematically depicted in Fig. 4c. a, Monolayer step edge geometry. We preserve periodic boundary conditions in the a₁-direction (the out-of-plane direction, with lattice spacing a₁), so that k₁ is a conserved crystal momentum. There are two step edges, one of type A and one of type B (see Supplementary Information), on each of the top and bottom surfaces. This is the minimal configuration of step edges that preserves inversion symmetry as well as periodic boundary conditions along the a₂-direction. b, Monolayer step edge dispersion with k₁ using the lattice shown in panel a. We only show the momentum range k₁∈ [0, π] because the spectrum in the range k₁∈ [π, 2π] is related by time-reversal symmetry. There is no nontrivial spectral flow (the spectrum is gapped). c, Local density of states (LDOS) for the A step edge. The LDOS is large for the low-energy bands closest to the gap, implying that they are well-localized at the step edge. d, Local density of states for the B step edge. This LDOS is small, implying that there are no low-energy states at the step edge. e, Bilayer step edge geometry. f, Bilayer step edge dispersion. There is still a gap, but the low-energy bands are slightly different from the monolayer case. g, Local density of states for the A step edge. The LDOS is again large for the low-energy bands, implying that they are well-localized at the same step edge as for the monolayer. h, Local density of states for the B step edge. Like the monolayer case, there are no step edge states close to the Fermi level.