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In this paper we systematically classify and describe bosonic symmetryprotected topological (SPT) phases in all physical spatial dimensions usingsemiclassical nonlinear Sigma model (NLSM) field theories. All the SPT phaseson a $d-$dimensional lattice discussed in this paper can be described by thesame NLSM, which is an O(d+2) NLSM in $(d+1)-$dimensional space-time, with atopological $\Theta-$term. The field in the NLSM is a semiclassical Landauorder parameter with a unit length constraint. The classification of SPT phasesdiscussed in this paper based on their NLSMs is consistent with the moremathematical classification based on group cohomology. Besides theclassification, the formalism used in this paper also allows us to explicitlydiscuss the physics at the boundary of the SPT phases, and it reveals therelation between SPT phases with different symmetries. For example, it givesmany of these SPT states a natural "decorated defect" construction.

Classification and Description of Bosonic Symmetry Protected Topological Phases with semiclassical Nonlinear Sigma models