QED2+1 in graphene: symmetries of Dirac equation in 2+1 dimensions

TL;DR

This paper analyzes the symmetry properties of the 2+1-dimensional Dirac equation in graphene, revealing a complex symmetry algebra and the emergence of a spin-like pseudospin from sublattice structure, connecting condensed matter physics with high energy phenomena.

Contribution

It provides a detailed symmetry analysis of the Dirac equation in graphene, highlighting the algebraic structure and the pseudospin's role as an angular momentum analogue.

Findings

Symmetry algebra is a direct sum of two gl(2,C) algebras plus an eight-dimensional abelian ideal.

Pseudospin acts as a real angular momentum despite the absence of electron spin.

The representation of the Dirac equation in graphene is reducible, affecting symmetry descriptions.

Abstract

It is well-known that the tight-binding Hamiltonian of graphene describes the low-energy excitations that appear to be massless chiral Dirac fermions. Thus, in the continuum limit one can analyze the crystal properties using the formalism of quantum electrodynamics in 2+1 dimensions (QED2+1) which provides the opportunity to verify the high energy physics phenomena in the condensed matter system. We study the symmetry properties of 2+1-dimensional Dirac equation, both in the non-interacting case and in the case with constant uniform magnetic field included in the model. The maximal symmetry group of the massless Dirac equation is considered by putting it in the Jordan block form and determining the algebra of operators leaving invariant the subspace of solutions. It is shown that the resulting symmetry operators expressed in terms of Dirac matrices cannot be described exclusively in terms of gamma matrices (and their products) entering the corresponding Dirac equation. It is a consequence of the reducibility of the considered representation in contrast to the 3+1-dimensional case. Symmetry algebra is demonstrated to be a direct sum of two gl(2,C) algebras plus an eight-dimensional abelian ideal. Since the matrix structure which determines the rotational symmetry has all required properties of the spin algebra, the pseudospin related to the sublattices (M. Mecklenburg and B. C. Regan, Phys. Rev. Lett. 106, 116803 (2011)) gains the character of the real angular momentum, although the degrees of freedom connected with the electron's spin are not included in the model. This seems to be graphene's analogue of the phenomenon called "spin from isospin" in high energy physics.