Abstract
In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang and Wen, the concept of equivalence classes of fermionic local unitary(FLU) transformations was proposed to systematically understand non-chiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensors (N^ij_k,F^ij_k,F^ijm,αβ_kln,χδ,n_i,d_i), which satisfy a set of nonlinear algebraic equations parameterized by phase factors Ξ^ijm,αβ_kl, Ξ^ij_kln,χδ, Ω^kim,αβ_jl and Ω^ki_jln,χδ. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-Yamagami category for Z_2N, which can be regarded as the Z_2N parafermion generalization of Ising fermionic topological phase.