Classification of composite Weyl fermions

Weyl nodes with chiral charge ±1 are classified as types I and type II according to the tilting of the conical band dispersion at the band degeneracy. Their Fermi surface, described by a quadratic form, is different for these two types. We extend this classification to the case of composite Weyl nodes with chiral charge larger than one. When the C₄ and C₆ rotation symmetries forbid the linear band dispersion on the plane perpendicular to the symmetry axis, new terms with quadratic and cubic momentum dependence must be included. Consequently, the Fermi surface produced by these band degeneracies are described by a 4 or 6 order algebraic surfaces. In this more complex situation, instead of classifying Fermi surfaces, we study numerically the possible Lifshitz transitions of the Fermi surface produced by a composite Weyl node as the chemical potential is varied. We use this methodology to study quadratic Weyl nodes generated by the C₄ rotation symmetry. We find that such band degeneracies present four different types of morphologies, two analogous to the type-I and type-II case of the conical Weyl nodes, and two new distinct morphologies within the type-II class. We illustrate the existence of these new types of band degeneracies in real materials such as bcc iron.