Reflectionless Klein tunneling of Dirac fermions: Comparison of split-operator and staggered-lattice discretization of the Dirac equation

TL;DR

This paper compares two discretization methods for the Dirac equation, showing that the split-operator approach preserves Klein tunneling better than the staggered lattice method due to fermion doubling issues.

Contribution

It demonstrates that the staggered lattice discretization introduces fermion doubling, leading to Klein tunneling breakdown, while the split-operator method effectively preserves the phenomenon.

Findings

Staggered lattice doubles the Brillouin zone size.

Fermion doubling causes Klein tunneling breakdown.

Split-operator approach preserves Klein tunneling.

Abstract

Massless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as "Klein tunneling", may be lost if the Dirac equation is discretized in space and time, because of scattering between multiple Dirac cones in the Brillouin zone. To avoid this, a staggered space-time lattice discretization has been developed in the literature, with one single Dirac cone in the Brillouin zone of the original square lattice. Here we show that the staggering doubles the size of the Brillouin zone, which actually contains two Dirac cones. We find that this fermion doubling causes a spurious breakdown of Klein tunneling, which can be avoided by an alternative single-cone discretization scheme based on a split-operator approach.