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 t(a), t(b), and t(c), respectively. For the isotropic case-namely, for t(a)=t(b)=t(c)-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 t(a), t(b), and t(c) are varied. It is shown that zero modes exist if parallel to parallel to t(b)/t(a)parallel to-1 parallel to <=parallel to t(c)/t(a)parallel to <=parallel to parallel to t(b)/t(a)parallel to+1 parallel to. The density of states near a zero mode is proportional to parallel to E parallel to but it is propotional to root parallel to E parallel to at the boundary of this condition