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

Abstract: Topological Dirac and Weyl semimetals have an energy spectrum that hosts Weyl nodes appearing in pairs of opposite chirality. Topological stability is ensured when the nodes are separated in momentum space and unique spectral and transport properties follow. In this work we study the effect of a space dependent Weyl node separation, which we interpret as an emergent background axial vector potential, on the electromagnetic response and the energy spectrum of Weyl and Dirac semimetals. This situation can arise in the solid state either from inhomogeneous strain or non-uniform magnetization and can also be engineered in cold-atomic systems. Using a semiclassical approach we show that 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. We then use two lattice models to analyze the effect of $\mathbf{B}5$ on the spectral properties of topological semimetals. We describe the emergent pseudo-Landau level structure for different spatial profiles of $\mathbf{B}_5$, revealing that (i) the celebrated surface states of Weyl semimetals, the Fermi arcs, can be reinterpreted as $n=0$ pseudo-Landau levels resulting from a $\mathbf{B}_5$ confined to the surface (ii) as a consequence of 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, which are the analogs of bound currents in magnetic materials. We conclude by discussing how our findings can be probed experimentally.