Confinement and Fermion Doubling Problem in Dirac-like Hamiltonians

TL;DR

This paper explores how confinement affects the fermion doubling problem in Dirac-like Hamiltonians, showing that certain boundary conditions can bypass the no-go theorem and demonstrating this with topological insulator and graphene nanoribbon models.

Contribution

It reveals that the symmetry breaking in Wilson's mass approach is equivalent to hard-wall boundary conditions, making the fermion doubling problem manageable under confinement.

Findings

Boundary conditions can eliminate fermion doublers in confined Dirac systems.

Transport properties in Bi$_2$Se$_3$ thin films are affected by confinement.

Band structures of zigzag graphene nanoribbons are consistent with the proposed approach.

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.