Role of the edges in a quasicrystalline Haldane model

TL;DR

This paper investigates how different edge configurations influence the topological properties and energy gaps in a quasicrystalline higher order topological insulator modeled by stacked Haldane layers with a twist.

Contribution

It reveals the critical role of edge types, especially bearded bonds, in gap formation near the charge neutrality point in a quasicrystalline topological insulator.

Findings

Gap opening depends on edge type and occurs at different energies.

Bearded bonds are essential for gap near charge neutrality.

Zigzag edges lead to symmetric gaps away from zero energy.

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 $30^\circ$ twist, whose structure is crystallographically 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.