Remarks on the tight-binding model of graphene

TL;DR

This paper examines the non-uniqueness of the tight-binding basis in graphene systems, emphasizing how operator forms depend on basis choice and can affect physical predictions, especially under disorder or magnetic fields.

Contribution

It clarifies the implications of basis choice in tight-binding models of graphene and analyzes operator forms in different bases for monolayer and bilayer graphene.

Findings

Operator forms vary with basis choice in graphene models.

Incorrect operator forms can lead to erroneous predictions.

Analysis applies to both monolayer and bilayer graphene.

Abstract

We address a simple but fundamental issue arising in the study of graphene, as well as of other systems that have a crystalline structure with more than one atom per unit cell. For these systems, the choice of the tight-binding basis is not unique. For monolayer graphene two bases are widely used in the literature. While the expectation values of operators describing physical quantities should be independent of basis, the form of the operators may depend on the basis, especially in the presence of disorder or of an applied magnetic field. Using the inappropriate form of certain operators may lead to erroneous physical predictions. We discuss the two bases used to describe monolayer graphene, as well as the form of the most commonly used operators in the two bases. We repeat our analysis for the case of bilayer graphene.