High-throughput screening for Weyl semimetals with S₄ symmetry

Based on irreducible representations (or symmetry eigenvalues) and compatibility relations (CR), a material can be predicted to be a topological/trivial insulator (satisfying CR) or a topological semimetal (violating CR). However, Weyl semimetals (WSMs) usually go beyond this symmetry-based strategy. In other words, Weyl nodes could emerge in a material, no matter if its occupied bands satisfy CR, or if the symmetry indicators are zero. In this work, we propose a new topological invariant χ for the systems with S₄ symmetry (i.e., the improper rotation S₄(≡IC_4z) is a proper fourfold rotation (C_4z) followed by inversion (I)), which can be used to diagnose the WSM phase. Moreover, χ can be easily computed through the one-dimensional Wilson-loop technique. By applying this method to the high-throughput screening in our first-principles calculations, we predict a lot of WSMs in both nonmagnetic and magnetic compounds. Various interesting properties (e.g., magnetic frustration effects, superconductivity and spin-glass order, etc.) are found in predicted WSMs, which provide realistic platforms for future experimental study of the interplay between Weyl fermions and other exotic states.