Zero modes of tight-binding electrons on the honeycomb lattice

Abstract

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 $\ensuremath{\mid}\ensuremath{\mid}\frac{{t}_{b}}{{t}_{a}}\ensuremath{\mid}\ensuremath{-}1\ensuremath{\mid}\ensuremath{\leqslant}\ensuremath{\mid}\frac{{t}_{c}}{{t}_{a}}\ensuremath{\mid}\ensuremath{\leqslant}\ensuremath{\mid}\ensuremath{\mid}\frac{{t}_{b}}{{t}_{a}}\ensuremath{\mid}+1\ensuremath{\mid}$. The density of states near a zero mode is proportional to $\ensuremath{\mid}E\ensuremath{\mid}$ but it is propotional to $\sqrt{\ensuremath{\mid}E\ensuremath{\mid}}$ at the boundary of this condition