Confinement and Fermion Doubling Problem in Dirac-like Hamiltonians

Abstract

We investigate the interplay between confinement and the fermion doubling problem in Dirac-like Hamiltonians. Individually, both features are well known. First, simple electrostatic gates do not confine electrons due to the Klein tunneling. Second, a typical lattice discretization of the first-order derivative $k \rightarrow -i\partial_x$ skips the central point and allow spurious low-energy, highly oscillating solutions known as fermion doublers. While a no-go theorem states that the doublers cannot be eliminated without artificially breaking a symmetry, here we show that the symmetry broken by the Wilson's mass approach is equivalent to the enforcement of hard-wall boundary conditions, thus making the no-go theorem irrelevant when confinement is foreseen. We illustrate our arguments by calculating the following: (i) the band structure and transport properties across thin films of the topological insulator Bi$_2$Se$_3$, for which we use ab-initio density functional theory calculations to justify the model; and (ii) the band structure of zigzag graphene nanoribbons.