Existence of three-fold Weyl points in generic admissible potentials

Weyl points are conically degenerate points on the spectral bands of three dimensional systems. In this paper, we investigate Weyl points for a large class of admissible potentials which are characterized only by certain symmetries. A rigorous justification for the existence of the three-fold Weyl points is provided in general three-dimensional problems for the first time. The existing results mainly focus on the weak potential or tight-binding regime. This paper provides a comprehensive proof of the existence in the generic regime. Our proof relies on the analytical structure of the energy bands and the corresponding Bloch modes in terms of the parameters. A renormalized determinant of the operator is used to characterize the energy. For the presence of local conical structures, we use the local analyticity of the corresponding eigenfunctions which are obtained through appropriate truncation. With the help of nice properties of analytical functions, we extend the existence of Weyl points from the weak potential limit to a generic regime. Our analysis lay a foundation for analyzing conically degenerate points in higher dimensions and with more symmetries. Some numerical simulations are provided to illustrate our analysis.