The question whether the mixed phase of a gapless superconductor can support a Landau level is a celebrated problem in the context of d-wave superconductivity, with a negative answer: the scattering of the subgap excitations (massless Dirac fermions) by the vortex lattice obscures the Landau level quantization. Here we show that the same question has a positive answer for a Weyl superconductor: the chirality of the Weyl fermions protects the zeroth Landau level by means of a topological index theorem. As a result, the heat conductance parallel to the magnetic field has the universal value G=1/2g_{0}Φ/Φ_{0}, with Φ as the magnetic flux through the system, Φ_{0} as the superconducting flux quantum, and g_{0} as the thermal conductance quantum.

In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schrödinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function u(r) is defined on a lattice as a solution of the differential equation H u(r) = 1, where H is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.

A 2π phase shift across a Josephson junction in a topological superconductor injects vortices into the chiral edge modes at opposite ends of the junction. When two vortices are fused they transfer charge into a metal contact. We calculate the time dependent current profile for the fusion process, which consists of ±e/2 charge pulses that flip sign if the world lines of the vortices are braided prior to the fusion. This is an electrical signature of the non-Abelian exchange of Majorana zero-modes.

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.

The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special ``staggered’’ grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form \bm{\mathcal H}\bm{\psi}=\bm{E}\bm{\mathcal P}\bm{\psi}ℋ𝛙=𝐄𝒫𝛙, with Hermitian tight-binding operators \bm{\mathcal H}ℋ, \bm{\mathcal P}𝒫, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.