Upper bound on the number of Weyl points born from a nongeneric degeneracy point

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Nongeneric isolated twofold degeneracy points, such as multi-Weyl points, split into Weyl points upon a generic perturbation that removes the fine tuning or protecting symmetry. The number of the resulting Weyl points is at least $|Q|$, where $Q$ is the topological charge associated to the nongeneric degeneracy point. Here, we show that such a nongeneric degeneracy point also has an birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. This upper bound is a local multiplicity associated to the nongeneric degeneracy point, an invariant of map germs known from singularity theory. This holds not only for the case of a three-dimensional parameter space with a Hermitian Hamiltonian, but also for the case of a two-dimensional parameter space with a chiral-symmetric Hamiltonian. We illustrate the power of this result for band structures of two- and three-dimensional crystals. Our work establishes a strong and powerful connection between singularity theory and topological band structures and, more broadly, parameter-dependent quantum systems.