Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases

We propose and prove a family of generalized Lieb-Schultz-Mattis (LSM) theorems for symmetry protected topological (SPT) phases on boson/spin models in any dimensions. The "conventional" LSM theorem, applicable to e.g. translation invariant system with an odd number spin-1/2 particles per unit cell, forbids symmetric short-range-entangled ground state such system. Here we focus systems no anomaly, where global/crystalline symmetries fractional spins within the cell ensure that SRE must be nontrivial SPT phase anomalous boundary excitations. Depending models, they can either strong or "higher-order" crystalline phases, characterized by surface/hinge/corner states. Furthermore, given group spatial assignment spins, are able determine all possible state, using real space construction based spectral sequence cohomology theory. provide examples one, two three dimensions, discuss physical realization these condensation excitations fractionalized phases.