Generalizing spin and pseudospin symmetries for relativistic spin 1/2 fermions

TL;DR

This paper generalizes the SU(2) spin and pseudospin symmetries in the Dirac equation, providing conditions on potentials to suppress matrix structures, with potential applications in systems like graphene.

Contribution

It extends the formalism of spin and pseudospin symmetries to more general Lorentz structures and potential configurations in various dimensions.

Findings

Derived conditions for Lorentz potentials to exhibit SU(2) symmetries.

Identified candidate potentials in different dimensions.

Proposed applications to physical systems such as graphene.

Abstract

We propose a generalization of pseudospin and spin symmetries, the SU(2) symmetries of Dirac equation with scalar and vector mean-field potentials originally found independently in the 70's by Smith and Tassie, and Bell and Ruegg. As relativistic symmetries, they have been extensively researched and applied to several physical systems for the last 18 years. The main feature of these symmetries is the suppression of the spin-orbit coupling either in the upper or lower components of the Dirac spinor, thereby turning the respective second-order equations into Schr\"odinger-like equations, i.e, without a matrix structure. In this paper we use the original formalism of Bell and Ruegg to derive general requirements for the Lorentz structures of potentials in order to have these SU(2) symmetries in the Dirac equation, again allowing for the suppression of the matrix structure of the second-order equation of either the upper or lower components of the Dirac spinor. Furthermore, we derive equivalent conditions for spin and pseudospin symmetries with 2- and 1-dimensional potentials and list some possible candidates for 3, 2, and 1 dimensions. We suggest applications for physical systems in three and two dimensions, namely electrons in graphene.