Birth Quota of Non-Generic Degeneracy Points

Abstract

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Non-generic isolated two-fold 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 non-generic degeneracy point. Here, we show that such a nongeneric degeneracy point also has a birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. The birth quota 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 electronic band structures of two-and three-dimensional crystals. Our work establishes a strong connection between singularity theory and topological band structures, and more broadly, parameter-dependent quantum systems.