Bifurcations of edge states -- topologically protected and non-protected -- in continuous 2D honeycomb structures

TL;DR

This paper investigates the bifurcation of topologically protected and non-protected edge states in continuous 2D honeycomb structures, highlighting conditions for their existence and behavior in various edge geometries and contrast regimes.

Contribution

The authors extend bifurcation theory for edge states in continuous honeycomb structures, distinguishing protected from non-protected bifurcations and analyzing their dependence on structure contrast and edge type.

Findings

Protected edge states exist at low contrast under specific Fourier coefficient conditions.

Non-protected edge states manifest as long-lived quasi-modes that leak energy.

Thresholds in medium contrast determine the existence of protected edge states for rational edges.

Abstract

This paper summarizes and extends the authors' work on the bifurcation of topologically protected edge states in continuous two-dimensional honeycomb structures. We consider a family of Schr\"odinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous two-dimensional honeycomb structures. The topologically protected bifurcation of edge states is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schr\"odinger operator. Numerical simulations for honeycomb structures of varying contrasts and "rational edges" (zigzag, armchair and others), support the following scenario: (a) For low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived {\it edge quasi-modes} which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge.