A classification of 2D fermionic and bosonic topological orders

TL;DR

The paper extends the Levin-Wen string-net and local unitary framework to interacting 2D fermionic systems by introducing fermionic local unitary (fLU) and generalized gfLU transformations. It encodes fermionic topological order data in the quadruple (N^{ij}_k, F^{ij}_k, F^{ijm,αβ}_{jkn,χδ}, d_i) subject to nonlinear consistency relations (Neq, Feq, Oeq, dFeq), and shows how fixed-point wavefunctions on graphs, along with F-, O-, and Y-moves, classify distinct orders. It also develops a categorical perspective via projective super fusion categories and provides concrete group-cohomology and group-supercohomology solutions to illustrate the construction. The work yields a path to constructing exact Hamiltonians for fermionic topological orders and broadens the mathematical framework beyond bosonic tensor categories to enriched, fermionic settings. Overall, it provides a systematic, constructive route to classify and realize 2+1D fermionic topological orders with gappable edges.

Abstract

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 2D bosonic systems. The two approaches reveal that the mathematical framework for 2+1D 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+1D fermionic topological orders with gappable edge in terms of the following set of data $(N^{ij}_k, F^{ij}_k, F^{ijm,αβ}_{jkn,χδ},d_i)$, that satisfy a set of non-linear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When $F^{ij}_k=0$, our result recovers the previous classification of 2+1D bosonic topological orders with gappable edge.

Figures (5)

• Figure 1: (Color online) (a) A fermion local unitary (gfLU) transformation $U_g$ acts in region A of a fermionic state $|\psi\>$ are formed by bosonic and fermionic operators in the region A. $U_g$ always contain even numbers of fermionic operators. (b) $U_g^\dag U_g=P$ is a projector, whose action does not change the state $|\psi\>$.
• 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 edges are labeled by $i,j,k,m,n=0,\cdots,N$. The state on the edges are labeled by $\al,\bt$. The vertices are labeled by $\u\al,\u\bt$. The $\u\al$ vertex has two incoming edges and $\al$ has a range $\al=1,\cdots,N^{kj}_n$. The $\u\bt$ vertex has two incoming edges and $\bt$ has a range $\bt=1,\cdots,N_{ij}^m$.
• Figure 3: (Color online) 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 $\v v$, hexagons by $\v p$, and links by $\v l$.
• Figure 5: $\cU_P$ is generated by an inverse H-move, an F-move, a dual H-move, an inverse F-moves and finally one O-move, which turns a hexagon graph into a tree graph. $(\cU_P)^\dag$ turns a tree graph into a hexagon graph.