  Gapless criteria that can efficiently determine whether a crystal is gapless
or not are particularly useful for identifying topological semimetals. In this
work, we propose a sufficient gapless criterion for three-dimensional
non-interacting crystals, based on the simplified expressions for the bulk
average value of the static axion field. The brief logic is that two different
simplified expressions give the same value in an insulator, and thus the
gapless phase can be detected by the mismatch of them. We demonstrate the
effectiveness of the gapless criterion in the magnetic systems with space
groups 26 and 13, where mirror, glide, and inversion symmetries provide the
simplified expressions. In particular, the gapless criterion can identify
gapless phases that are missed by the symmetry representation approach, as
illustrated by space group 26. Our proposal serves as a guiding principle for
future discovery of topological semimetals.
