Persistence of Zero Landau Energies Protected from Vorticities of Opposite Chiral Weyl Nodes

The zero energy Landau level is a topological property of the Weyl nodes with linear dispersion. The Landau levels can change when various nearby Weyl nodes interact with each other through the coupling induced by the magnetic field. It is expected that these zero Landau levels may disappear under high field when the nodes are of opposite chiralities. We demonstrate that even for systems with opposite chiral Weyl nodes, different symmetry requirement can result in different conditions of these Landau levels. The chiralities of the Weyl nodes themselves are not enough to decide whether these zero energies can persist. It is the vorticities projected on the plane perpendicular to the magnetic field that decide the either survival or gaping out of the zero energies. Different ways of nodes connection required by the symmetries produce different number of vorticities on the plane. The gaping out of the zero energy levels turning the system into an insulator has no room for transport, while the persistence of zero energies still allows the chiral anomaly to transport electrons along the magnetic field direction.