Author(s):

Authors: C. L. Fefferman, J. P. Lee-Thorp, M. I. Weinstein

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.

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