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