Boundary conditions for the two-dimensional fermions in ribbons of the
hexagonal lattice are studied in the dice model whose energy spectrum in
infinite system consists of three bands with one completely flat band of zero
energy. Like in graphene the regular lattice terminations are of the armchair
and zigzag types. However, there are four possible zigzag edge terminations in
contrast to graphene where only one type of zigzag termination is possible.
Determining the boundary conditions for these lattice terminations, the energy
spectra of pseudospin-1 fermions in dice model ribbons with zigzag and armchair
boundary conditions are found. It is shown that the energy levels for armchair
ribbons display the same features as in graphene except the zero energy flat
band inherent to the dice model. In addition, unlike graphene, there are no
propagating edge states localized at zigzag boundary and there are specific
zigzag terminations which give rise to bulk modes of a metallic type in dice
model ribbons. We find that the existence of the flat zero-energy band in the
dice model is very robust and is not affected by the zigzag and armchair
boundaries.Comment: 16 pages, 7 figure
