Figure 3

Zeeman coupling and Landau level spectrum. (a) Cartoon illustration of two doubly-degenerate bands crossing. The four-fold degenerate crossing point describes Dirac fermion quasiparticles. (b) In the presence of a Zeeman field, the two Dirac fermions described in (a) split into 2 pairs of Weyl fermions (i.e. \(\to {W}_{1}^{+}+{W}_{1}^{-}\)). All pairs of generated Weyl fermions arise from the crossing between two singly-degenerate bands (red and yellow). The blue shaded region corresponds to section in the Brillouin zone with a non-zero Chern number. (c) Cartoon illustration of a doubly-degenerate band (blue) crossing a singly-degenerate band to create a pair of triply-degenerate fermions at the crossing points. (d) In the presence of a magnetic field, the doubly-degenerate band splits into two singly-degenerate bands, which then cross the singly-degenerate band at two different locations. Because the pair of generated Weyl fermions are not the crossing points between the same two singly-degenerate bands, the blue shaded area with a non-zero Chern number is distinctly different from the Dirac fermion case presented in (b). (e,f) A cartoon schematic for the splitting of a triply-degenerate type-I and type-II fermion (purple sphere) in the absence (left panels) and presence (right panels) of a \({ {\mathcal M} }_{z}\) field, respectively. (g,i) Landau level spectrum for type-I (left panel) and type-II (right panel) Weyl fermions, respectively, for a magnetic field along the z direction. Type-I Weyl fermions produce a gapless chiral Landau level spectrum, and realize the chiral anomaly of quantum field theory. Type-II Weyl fermions have a gapless chiral Landau level spectrum only when the magnetic field points along certain directions, and, therefore, realize an anisotropic chiral anomaly. (h,j) Landau level spectrum for Class α triply-degenerate fermions of type-I (h) and type-II (j), respectively. The zeroth order Landau level band is in red.