Zero modes and the edge states of the honeycomb lattice

TL;DR

This paper investigates edge states in the honeycomb lattice with various boundary conditions, revealing new anisotropy-induced states and classifying different types of localized and extended states.

Contribution

It provides analytical solutions for edge states in honeycomb lattices with multiple edge types, including newly identified states in anisotropic armchair configurations.

Findings

Edge states are analytically obtained for zigzag, bearded, and mixed edges.

Newly discovered edge states exist in anisotropic armchair edges.

The study classifies localized, extended, and degenerate states in the system.

Abstract

The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states and states with macroscopic degeneracy.