Classification and description of bosonic symmetry protected topological phases with semiclassical nonlinear sigma models

In this paper, we systematically classify and describe bosonic symmetry protected topological (SPT) phases in all physical spatial dimensions using semiclassical nonlinear sigma model (NLSM) field theories. All the SPT phases on a $d\text{-dimensional}$ lattice discussed in this paper can be described by the same NLSM, which is an O($d+2$) NLSM in $(d+1)\text{-dimensional}$ space-time, with a topological $\Theta$ term. The field in the NLSM is a semiclassical Landau order parameter with a unit length constraint. The classification of SPT phases discussed in this paper based on their NLSMs is Completely Identical to the more mathematical classification based on group cohomology given in X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013) and Science 338, 1604 (2012). Besides the classification, the formalism used in this paper also allows us to explicitly discuss the physics at the boundary of the SPT phases, and it reveals the relation between SPT phases with different symmetries. For example, it gives many of these SPT states a natural “decorated defect” construction.