Classification of two-dimensional fermionic and bosonic topological orders

The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in two-dimensional (2D) bosonic systems. The two approaches reveal that the mathematical framework for (2 + 1)-dimensional (2 + 1)D bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of (2 + 1)D fermionic topological orders with gappable edge in terms of the following set of data (N-k(ij), F-k(ij), F-jkn,chi delta(ijm,alpha beta), d(i)), which satisfy a set of nonlinear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When F-k(ij) = 0, our result recovers the previous classification of 2 + 1D bosonic topological orders with gappable edge.