Zero modes, energy gap, and edge states of anisotropic honeycomb lattice in a magnetic field

TL;DR

This paper investigates how anisotropy in hopping integrals affects zero modes, energy gaps, and edge states in a honeycomb lattice under a magnetic field, revealing conditions for zero modes and the impact of anisotropy on edge states.

Contribution

It analytically derives conditions for zero modes and edge states in an anisotropic honeycomb lattice in a magnetic field, highlighting the effects of anisotropy on energy gaps and edge state behavior.

Findings

Tiny anisotropy opens a gap around zero energy in a magnetic field.

The energy gap exhibits non-perturbative, exponential dependence on magnetic field.

Anisotropy causes abrupt changes in the number of zero energy edge states depending on edge shape.

Abstract

We present systematic study of zero modes and gaps by introducing effects of anisotropy of hopping integrals for a tight-binding model on the honeycomb lattice in a magnetic field. The condition for the existence of zero modes is analytically derived. From the condition, it is found that a tiny anisotropy for graphene is sufficient to open a gap around zero energy in a magnetic field. This gap behaves as a non-perturbative and exponential form as a function of the magnetic field. The non-analytic behavior with respect to the magnetic field can be understood as tunneling effects between energy levels around two Dirac zero modes appearing in the honeycomb lattice, and an explicit form of the gap around zero energy is obtained by the WKB method near the merging point of these Dirac zero modes. Effects of the anisotropy for the honeycomb lattices with boundaries are also studied. The condition for the existence of zero energy edge states in a magnetic field is analytically derived. On the basis of the condition, it is recognized that anisotropy of the hopping integrals induces abrupt changes of the number of zero energy edge states, which depend on the shapes of the edges sensitively.