Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions

Symmetry-protected topological (SPT) phases in two-dimensions can be largely described by Chern-Simons topological field theories. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We also apply the theory to fermionic SPT phases with $Z_m$ symmetry and find the classification of SPT phases depends on the parity of $m$: for even $m$ there are $2m$ classes, $m$ out of which is intrinsically fermionic SPT phases and can not be realized in any bosonic system. We outline the general classification scheme for fermionic SPT phases in two dimensions.