Zero modes of tight binding electrons on the honeycomb lattice

TL;DR

This paper investigates the conditions for zero energy modes in tight binding electrons on a honeycomb lattice with anisotropic hopping parameters, revealing how these modes depend on hopping ratios and their impact on the density of states.

Contribution

It provides a detailed analysis of zero modes in anisotropic honeycomb lattices, deriving conditions for their existence and characterizing the density of states near these modes.

Findings

Zero modes exist when specific ratios of hopping parameters satisfy a certain inequality.

The position of zero modes shifts in momentum space as hopping parameters vary.

Density of states near zero modes scales linearly with energy, changing at boundary conditions.

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 $||\frac{t_b}{t_a}| -1| \leq |\frac{t_c}{t_a}| \leq ||\frac{t_b}{t_a}|+1|$. The density of states near a zero mode is proportional to $|E|$ but it is propotional to $\sqrt{|E|}$ at the boundary of this condition.