  Based on irreducible representations (or symmetry eigenvalues) and
compatibility relations, a material can be predicted to be a
topological/trivial insulator [satisfying compatibility relations] or a
topological semimetal [violating compatibility relations]. However, Weyl
semimetals usually go beyond this symmetry-based strategy. In other words, Weyl
nodes could emerge in a material, no matter if its occupied bands satisfy
compatibility relations, or if the symmetry indicators are zero. In this work,
we propose a new topological invariant $\chi$ for the systems with S$_4$
symmetry [i.e., the improper rotation S$_4$ ($\equiv$ IC$_{4z}$) is a proper
four-fold rotation (C$_{4z}$) followed by inversion (I)], which can be used to
diagnose the Weyl semimetal phase. Moreover, $\chi$ can be easily computed
through the one-dimensional Wilson-loop technique. By applying this method to
the high-throughput screening in first-principles calculations, we predict a
lot of Weyl semimetals in both nonmagnetic and magnetic compounds. Various
interesting properties (e.g. magnetic frustration effects, superconductivity
and spin-glass order, etc.) are found in predicted Weyl semimetals, which
provide realistic platforms for future experimental study of the interplay
between Weyl fermions and other exotic states.
