It is well-known that the dynamics of low energy electron in graphene honeycomb lattice near the K/K points can be described, in tight-binding approximation, by 2+1 massless Dirac equation. Graphenes spin equivalent, pseudospin, arises from the degen
eracy introduced by the honeycomb lattices two inequivalent atomic sites per unit cell. Mecklenburg and Regan (Phys. Rev. Lett. 106 (2011), 116803) have shown that, contrary to the common view, the pseudospin has all attributes of real angular momentum. In some circumstances, the internal symmetries can produce an important contribution to angular momentum. This phenomenon has been known for many years in particle physics and called spin from isospin. We show that similar mechanism works in the case of lattice pseudospin.
Nonrelativistic conformal groups, indexed by l=N/2, are analyzed. Under the assumption that the mass parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown t
hat the corresponding dynamical system describes, within Ostrogradski framework, the nonrelativistic particle obeying (N+1)-th order equation of motion. As a special case, the Schroedinger group and the standard Newton equations are obtained for N=1 (l=1/2).
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 electrons spin are not included in the model. This seems to be graphenes analogue of the phenomenon called spin from isospin in high energy physics.
