Time-reversal-invariant Weyl semimetals indicated by symmetry indictors and invariants

For the time-reversal(TR)-breaking centrosymmetric systems, an odd number of all the even/odd parity occupied bands, at eight inversion-symmetry-invariant (ISI) momenta, indicates the appearance of Weyl points in the systems. Here, based on the the first-principles calculations and symmetry analysis, we demonstrate that, for the TR-preversing noncentrosymmetric systems with S4 symmetry, the Weyl semimetal phase can be characterised by the nonequality between an invariant $\eta$ and an S4 indicator $z_2$. By applying it, we find that some candicates can be ideal Weyl semimetals in a noncentrosymmetric space group with S4 symmetry. Our first-principles calculations show that four pairs of Weyl points are located in the kx,y = 0 planes, being right at the charge neutrality level. An effective model has been built and captures the nontrivial topology in these materials. Our strategy to find the Weyl points by using symmetry indicators and invariants opens a new route to search for Weyl semimetals in TR-invariant systems.