A classification of 2D fermionic and bosonic topological orders

TL;DR

This paper extends the classification framework of 2D topological orders from bosonic to fermionic systems, providing a comprehensive set of algebraic data and exactly solvable models for fermionic topological phases with gappable edges.

Contribution

It introduces a new classification scheme for 2+1D fermionic topological orders using algebraic data, generalizing previous bosonic classifications and enabling construction of exact Hamiltonians.

Findings

Classifies 2D fermionic topological orders with gappable edges.

Recovers bosonic classification when fermionic data is zero.

Provides a method to construct exactly solvable Hamiltonians.

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 $(N^{ij}_k, F^{ij}_k, F^{ijm,\alpha\beta}_{jkn,\chi\delta},d_i)$, 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.