Form-preserving Darboux transformations for $4\times 4$ Dirac equations

TL;DR

This paper introduces a reducible Darboux transformation method for $4\times4$ Dirac equations that preserves physical interaction forms, enabling the construction of exactly solvable, reflectionless models relevant to graphene physics.

Contribution

It develops a reducible Darboux transformation framework that maintains physical interaction forms in $4\times4$ Dirac systems, facilitating solvable models with desired properties.

Findings

Constructed reflectionless models for Dirac fermions in graphene.

Demonstrated preservation of physical interactions via reducible Darboux transformations.

Provided explicit examples of solvable systems with no backscattering.

Abstract

Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by $4\times4$ Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scattering and spin-orbit interaction of Dirac fermions in graphene. We use the reducible Darboux transformation to construct exactly solvable models of these systems where backscattering is absent, i.e. the models are reflectionless.