Lieb-Schultz-Mattis constraints in 3D: application to the pyrochlore and the diamond lattices

Lieb-Schultz-Mattis theorems are a powerful family of constraints restricting the types of ground states of a lattice magnet. Two formalisms have been established: one uses the notation of lattice homotopy [Po et al., PRL 119, 127202 (2017)] and the other uses the idea of quantum anomalies [Else and Thorngren, PRB 101 224437, (2020)]. Here we discuss how these seemingly unrelated formalisms are related, and how these ideas can be employed to obtain a set of complete LSM constraints in three-dimensional lattice magnets. We will focus on the LSM theorems and filling constraints on the pyrochlore, diamond, and breathing pyrochlore lattices, which host some of the prototypical spin liquids in three dimensions. We go beyond the existing notation of classification by giving a detailed analysis of the microscopic origin for the nontrivial topological classes. This analysis allows us to obtain and track the trivialization of these anomalies under the breaking of lattice symmetries. Finally, we discuss the matching of the LSM anomalies and anomalies in quantum spin liquids. The principles and calculation methods presented in this work can be applied to all 3D lattice magnets.