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

We propose and proof a generalized Lieb-Schultz-Mattis (LSM) theorem for symmetry protected topological (SPT) phases on boson/spin models in any dimensions. The "conventional'' LSM theorem, e.g. spin-1/2 per unit cell on square lattice, disallows symmetric short-range-entangled (SRE) phase. Here, we focus on systems with fractional spins, but have no LSM anomaly. Thus, it is possible to have symmetric SRE phases in the long wavelength. We show that, symmetric SRE phases obtained in these systems must be nontrivial SPT phases of both on-site and spatial symmetries. Depending on models, they can be either strong or higher-order SPT phases, characterized by nontrivial edge/corner states. Furthermore, given global symmetry group and fractional spins, we are able to determine all possible SPT phases by using a spectral sequence expansion of group cohomology. We also provide examples in various dimensions, and discuss possible physical realization of these SPT phases based on topological defects/quasiparticles condensation picture.