Quantum oscillations in Weyl semimetals: A surface theory approach

Weyl semimetals are characterized by an even number of bulk Weyl points and surface Fermi arcs connecting their surface projections. It has been shown that when a magnetic field is applied perpendicular to the surface, an unusual kind of quantum oscillations appear. Semiclassically, these oscillations are associated with electron motion in momentum space on the surface Fermi arcs as well as tunneling between the top and bottom surfaces through the bulk chiral Landau level. In this work, we develop an effective surface theory for Weyl semimetals by integrating out the bulk, leaving only the top and bottom infinite surfaces. We use our model to analyze the Landau level spectrum and compare the results with a full three-dimensional calculation as well as the semiclassical theory. We find that the surface theory is a good approximation for the three-dimensional model and agrees with the semiclassical analysis whenever the Weyl points are far from each other. We analytically analyze the case of overlapping Weyl points with short Fermi arcs and find an energy gap which appears as a phase shift in the quantum oscillations, in agreement with our numerics. We also study the case of a type-II Weyl semimetal and show that the absence of the bulk chiral Landau level prevents the formation of surface Landau levels.