The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space-time lattice that removes both obstructions is introduced. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single-cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time-evolution operator is discontinuous at Brillouin zone boundaries.

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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.

We calculate the Landau levels of a Kramers–Weyl semimetal thin slab in a perpendicular magnetic field B. The coupling of Fermi arcs on opposite surfaces broadens the Landau levels with a band width that oscillates periodically in 1/B. We interpret the spectrum in terms of a one-dimensional superlattice induced by magnetic breakdown at Weyl points. The band width oscillations may be observed as 1/B-periodic magnetoconductance oscillations, at weaker fields and higher temperatures than the Shubnikov–de Haas oscillations due to Landau level quantization. No such spectrum appears in a generic Weyl semimetal, the Kramers degeneracy at time-reversally invariant momenta is essential.

Methods to discretize the Hamiltonian of a topological insulator or topological superconductor, without giving up on the topological protection of the massless excitations (respectively, Dirac fermions or Majorana fermions) are reviewed. The method of tangent fermions, pioneered by Richard Stacey, is singled out as being uniquely suited for this purpose. Tangent fermions propagate on a 2+1${2\bm {+}1}$ dimensional space‐time lattice with a tangent dispersion: tan2(bold-italicε/2)=tan2(kx/2)+tan2(ky/2)${\text{tan}^2 (\bm {\varepsilon }/2) \bm {=} \text{tan}^2 (k_x/2) \bm {+}\text{tan}^2 (k_y/2)}$ in dimensionless units. They avoid the fermion doubling lattice artefact that will spoil the topological protection, while preserving the fundamental symmetries of the Dirac Hamiltonian. Although the discretized Hamiltonian is nonlocal, as required by the fermion‐doubling no‐go theorem, it is possible to transform the wave equation into a generalized eigenproblem that is local in space and time. Applications that are discussed include Klein tunneling of Dirac fermions through a potential barrier, the absence of localization by disorder, the anomalous quantum Hall effect in a magnetic field, and the thermal metal of Majorana fermions.