Strained and inhomogeneous Weyl and Dirac semimetals: Transport in axial magnetic fields and Fermi arc surface states from pseudo Landau levels

Dirac and Weyl semimetals host topologically stable Weyl nodes appearing in pairs of opposite chirality. In this work we allow a space-time dependent Weyl node separation, which acts as a background axial vector potential on the electromagnetic response and the energy spectrum of these materials. This situation arises either from inhomogeneous strain, non-uniform magnetization and also in cold-atomic systems. The resulting axial magnetic field $\mathbf{B}_{5}$ is observable through an enhancement of the conductivity as $\sigma\sim \mathbf{B}_{5} ^{2}$ due to an underlying chiral pseudo magnetic effect. Using two lattice models, we analyze the effect of $\mathbf{B}_5$ on the spectral properties of topological semimetals, revealing that (i) the surface Fermi arcs, can be reinterpreted as $n=0$ pseudo-Landau levels resulting from a $\mathbf{B}_5$ confined to the surface (ii) position-momentum locking a bulk $\mathbf{B}_5$ creates pseudo-Landau levels interpolating in real space between Fermi arcs at opposite surfaces and (iii) there are equilibrium bound currents proportional to $\mathbf{B}_{5}$ that average to zero over the sample, analogs of bound currents in magnetic materials. We conclude by discussing how our findings cour findings can be probed experimentally.