The Lieb-Schultz-Mattis-type filling constraints in the 1651 magnetic space groups

We present the first systematic study of the filling constraints to realize a `trivial’ insulator symmetric under magnetic space group $\mathcal{M}$. The filling $\nu$ must be an integer multiple of $m^{\mathcal{M}}$ to avoid spontaneous symmetry breaking or fractionalization in gapped phases. We improve the value of $m^{\mathcal{M}}$ in the literature and prove the tightness of the constraint for the majority of magnetic space groups. The result may shed light on the material search of exotic magnets with fractionalization.