Lieb–Schultz–Mattis theorem and the filling constraint

Following recent developments in the classification of bosonic short-range entangled phases, we examine many-body quantum systems whose ground state fractionalization obeys Lieb-Schultz-Mattis (LSM) theorem. We generalize topological such phases by LSM anomalies (arXiv:1907.08204) to take magnetic and non-symmorphic lattice effects into account, provide direct computations anomaly specific examples. show that anomaly-free condition coincides with established filling constraints (arXiv:1705.09298, arXiv:1505.04193), also derived new ones on novel crystalline systems.