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

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.

Tangent Fermions: Dirac or Majorana Fermions on a Lattice Without Fermion Doubling

“…Klein tunneling of tangent fermions was studied in ref. [36], based on a space‐time lattice generalization [ 32 ] of the generalized eigenproblem (Equation (22)). The stationary equation $${\cal H}\Psi =E{\cal P}\Psi$$ can be converted into a time‐dependent equation (time step $$\delta t$$) upon substitution of Ψ on the left‐hand‐side by $$\tfrac{1}{2}[\Psi (t+\delta t)+\Psi (t)]$$, and of $$E\Psi$$ on the right‐hand‐side by $$(i\hbar /\delta t)[\Psi (t+\delta t)-\Psi (t)]$$.…”

“…Klein tunneling of tangent fermions was studied in ref. [36], based on a space-time lattice generalization [32] of the generalized eigenproblem (Equation ( 22)). The stationary equation Ψ = EΨ can be converted into a time-dependent equation (time step 𝛿t) upon substitution of Ψ on the left-hand-side by 1 2 [Ψ(t + 𝛿t) + Ψ(t)], and of EΨ on the right-hand-side by (iℏ∕𝛿t…”

“…Full transmission is obtained for the tangent discretization without fermion doubling (left panel), while fermion doubling causes reflections for a staggered space-time lattice discretization[38] (right panel). Reproduced with permission [36]. Copyright 2022, IOP Publishing.…”