Dirac fermions in graphene using the position-dependent translation operator formalism

Abstract

Within the position-dependent translation operator formalism for quantum systems, we obtain analytical expressions for the eigenstates and the Landau level spectrum of Dirac fermions in graphene in the presence of a perpendicularly applied magnetic field and, as a consequence of the formalism, with a generalized form of the momentum operator. Moreover, we explore the behavior of wave packet dynamics in such a system by considering different initial pseudospin polarizations and metric parameters. Our findings show that the Landau levels, the wave packet trajectories, and velocities are significantly affected by the choice of the metric in the non-Euclidean space of the deformed momentum operator, exhibiting a tunable energy level spacing. In the dynamics analysis, one observes an enhancement of the oscillation amplitude of the average positions for all investigated pseudospin polarizations due to the nonsymmetric evolution of the wave packet induced by the different metrics in the system. The present formalism is shown to be a theoretical platform to describe the effects of two scenarios due to (i) a lattice deformation in graphene, giving rise to a natural Fermi velocity renormalization, or even (ii) a nonuniform mass term, induced by a specific substrate, that varies on a length scale much greater than the magnetic field length.