Classification of symmetry-protected phases for interacting fermions in two dimensions

Recently, it has been established that two-dimensional bosonic symmetry-protected topological (SPT) phases with on-site unitary symmetry G can be completely classified by the group cohomology H3(G,U(1)). Later, group supercohomology was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the algebraic theory of symmetry and defects in two-dimensional topological phases. We reproduce the partial classifications given by group supercohomology, and we also show that with an additional H1(G,Z2) structure, a complete classification of SPT phases for two-dimensional interacting fermion systems with a total symmetry group G×Z2f is obtained. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.