Birth Quota of Non-Generic Degeneracy Points

TL;DR

This paper introduces the concept of a birth quota for non-generic degeneracy points in Hamiltonians, establishing an invariant that limits the number of Weyl points that can emerge from such degeneracies.

Contribution

It connects singularity theory with topological band structures by defining a local multiplicity invariant for non-generic degeneracy points in quantum systems.

Findings

Defines the birth quota as an invariant of map germs.

Shows the birth quota applies to both Hermitian and chiral-symmetric Hamiltonians.

Demonstrates the concept with band structures in 2D and 3D crystals.

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 non-generic 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 non-generic 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.