Singularity theory of Weyl-point creation and annihilation

Weyl points (WP) are robust spectral degeneracies, which can not be split by small perturbations, as they are protected by their non-zero topological charge. For larger perturbations, WPs can disappear via pairwise annihilation, where two oppositely charged WPs merge, and the resulting neutral degeneracy disappears. The neutral degeneracy is unstable, meaning that it requires the fine-tuning of the perturbation. Fine-tuning of more than one parameter can lead to more exotic WP mergers. In this work, we reveal and analyze a fundamental connection of the WP mergers and singularity theory: phase boundary points of Weyl phase diagrams, i.e., control parameter values where Weyl point mergers happen, can be classified according to singularity classes of maps between manifolds of equal dimension. We demonstrate this connection on a Weyl--Josephson circuit where the merger of 4 WPs draw a swallowtail singularity, and in a random BdG Hamiltonian which reveal a rich pattern of fold lines and cusp points. Our results predict universal geometrical features of Weyl phase diagrams, and generalize naturally to creation and annihilation of Weyl points in electronic (phononic, magnonic, photonic, etc) band-structure models, where Weyl phase transitions can be triggered by control parameters such as mechanical strain.