A Block-diagonal form for four-component operators describing Graphene Quantum Dots

TL;DR

This paper analyzes four-component Dirac operators modeling graphene quantum dots, showing they can be reduced to two two-component operators under certain conditions, thus extending known results and connecting continuum models to tight-binding boundary conditions.

Contribution

It introduces a block-diagonal form for four-component Dirac operators with specific boundary conditions, linking continuum and tight-binding models of graphene quantum dots.

Findings

Hamiltonian is unitarily equivalent to two two-component operators for constant boundary parameters.

Identifies boundary conditions from tight-binding models that produce block-diagonal operators.

Extends known results from two-component to four-component Dirac operators.

Abstract

We consider four-component Dirac operators on domains in the plane. With suitable boundary conditions, these operators describe graphene quantum dots. The most general boundary conditions are defined by a matrix depending on four real parameters. For operators with constant boundary parameters we show that the Hamiltonian is unitary equivalent to two copies of the two-component operator. This allows to extend the known results for this type of operators to the four-component case. As an application, we identify the boundary conditions from the tight-binding model for graphene that give rise to a block-diagonal operator in the continuum limit.