Landau Levels of Double-Weyl Nodes in a Simple Lattice Model

Abstract

In the Weyl semimetals, a recently discovered class of bulk materials, inverted band gap closes in the first Brillouin zone at topologically protected points of degeneracy called the Weyl nodes. By using the Chern number formalism it is possible to assign to each of the nodes an integer topological charge Q. While around typical Weyl points the energy disperses linearly in all three directions, double-Weyl nodes (with |Q| = 2) exhibit quadratic dispersion in two directions and linear in the third one. We use a simple 2-band tight-binding lattice model to investigate the dispersion of the Landau levels in the presence of quantizing magnetic field in the vicinity of a double-Weyl node. In the long wavelength limit we obtain analytically the expected presence of two chiral levels. In addition, we find numerous level crossings between the non-chiral Landau levels and the chiral ones, a feature which is distinct from the single node case. Calculations for a finite-size sample, both with periodic and hard-wall boundary conditions (the latter corresponding to slab geometry), show that the two chiral levels hybridize in the conduction band with the two lowest non-chiral Landau levels. In the case of slab geometry these four levels are responsible for the formation of a protected surface state.