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 $\left(2+1\right)\mathrm{D}$ 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 $(2+1)\mathrm{D}$ 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 $2+1\mathrm{D}$ bosonic topological orders with gappable edge.

Figure 1

(a) A fermion local unitary (gfLU) transformation $U_g$ acts in region A of a fermionic state; $|\psi \rangle$ are formed by bosonic and fermionic operators in the region A. $U_g$ always contains even numbers of fermionic operators. (b) $U_g^{†}{U}_g=P$ is a projector whose action does not change the state $|\psi \rangle$.

Figure 2

The lattice is a graph (described by the blue lines) with branching structure. The black lines describe the dual lattice. The state on the vertices are labeled by $i,j,k,m,n=0,...,N$. The state on the vertices are labeled by $\alpha ,\beta$. The vertices are labeled by $\underline{\alpha },\underline{\beta }$. The $\underline{\alpha }$ vertex has two incoming edges and $\alpha$ has a range $\alpha =1,...,N_n^{kj}$. The $\underline{\beta }$ vertex has two outgoing edges and $\beta$ has a range $\beta =1,...,N_{ij}^m$.

Figure 3

Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.

Figure 4

A honeycomb lattice. The vertices are labeled by $\mathit{v}$, hexagons by $\mathit{p}$, and links by $\mathit{l}$.

Figure 5

${\mathcal{U}}_P$ is generated by an inverse H-move, an F-move, a dual H-move, an inverse F-move, and finally one O-move, which turns a hexagon graph into a tree graph. ${\left({\mathcal{U}}_P\right)}^{†}$ turns a tree graph into a hexagon graph.