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Gapless criteria that can efficiently determine whether a crystal is gaplessor not are particularly useful for identifying topological semimetals. In thiswork, we propose a sufficient gapless criterion for three-dimensionalnon-interacting crystals, based on the simplified expressions for the bulkaverage value of the static axion field. The brief logic is that two differentsimplified expressions give the same value in an insulator, and thus thegapless phase can be detected by the mismatch of them. We demonstrate theeffectiveness of the gapless criterion in the magnetic systems with spacegroups 26 and 13, where mirror, glide, and inversion symmetries provide thesimplified expressions. In particular, the gapless criterion can identifygapless phases that are missed by the symmetry representation approach, asillustrated by space group 26. Our proposal serves as a guiding principle forfuture discovery of topological semimetals.

Gapless Criterion for Crystals from Effective Axion Field