Role of the edges in a quasicrystalline Haldane model

Abstract

We study the role of the edges in determining the features of the topological phase in a quasicrystalline higher order topological insulator. We consider a specific model consisting of two stacked Haldane models with opposite Chern number and a 30 • twist, whose structure is cristallographically equivalent to that of the graphene quasicrystal. We find that the gap-opening in the low energy spectrum of the higher order topological insulator occurs at different energies when different kinds of edges are considered. Crucially, bearded bonds appear to be necessary for the gap to appear close to the charge neutrality point. In the more realistic case of zigzag edges, the gap opens symmetrically in the electron and hole sectors, away from zero energy. We explain our findings by inspecting the edge-bands of the decoupled bilayer, in the approximation of quasi-periodicity.

Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator

“…We then crop our lattice with a square shape centered in the rotation center, and oriented in such a way that the edges are not jagged. As already noted in [72], the edges of the resulting system-which possesses C 4 global rotational symmetry -can only be given by two possible pairs of the upper and lower layer edges: bearded-armchair or zigzagarmchair. However, even once the combination of the edges has been chosen, there are still two families of square lattices that can be obtained: these differ with each other for the disposition of the lattice sites around the vertices 1 .…”

“…The result is that the system, thanks to the presence of the rotational symmetry, hosts topological bound states, which are degenerate, isolated in energy from the others, and localized at the vertices of the structure. A subsequent work [72] showed that these properties also characterize samples with armchair-zigzag edges. However, in this case, the energy at which the gap opens drastically changes.…”

“…In [72], it was shown how the properties of the model for a HOTI on the (spinless) graphene quasicrystal first proposed in [71], depend on the shape of the edges. Here we want to discuss how, even when the edge combination is fixed, the features of the HOTI phase can still depend on the specific configuration of the sites near the vertices.…”

“…In this case, the gap does not open at the charge neutrality point[72].Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 July 2022 doi:10.20944/preprints202207.0437.v1…”

Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator

“…The system is hence a higher order topological insulator with d = 2 and n = 2. A subsequent work [72] showed that these properties also characterize samples with armchairzigzag edges. However, in this case, the energy at which the gap opens drastically changes.…”