  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.
