Zero modes of tight-binding electrons on the honeycomb lattice

Tight-binding electrons on the honeycomb lattice are studied where nearest-neighbor hoppings in the three directions are tₐ, tb, and tc, respectively. For the isotropic case---namely, for tₐ=tb=tc---two zero modes exist where the energy dispersions at the vanishing points are linear in momentum k. Positions of zero modes move in the momentum space as tₐ, tb, and tc are varied. It is shown that zero modes exist if ${}{}{{t}b}{{t}ₐ}{}{-}1{}{}{}{{t}c}{{t}ₐ}{}{}{}{}{{t}b}{{t}ₐ}{}+1{}$. The density of states near a zero mode is proportional to ${}E{}$ but it is propotional to $√{{}E{}}$ at the boundary of this condition