A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries

Abstract

We propose that, up to invertible topological orders, 2+1D fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$ are classified by non-degenerate unitary braided fusion categories (UBFC) over a symmetric fusion category (SFC); the SFC describes a fermionic product state without symmetry or a fermionic/bosonic product state with symmetry $G$, and the UBFC has a modular extension. We developed a simplified theory of non-degenerate UBFC over a SFC based on the fusion coefficients $N_k^{ij}$ and spins $s_i$. This allows us to obtain a list that contains all 2+1D fermionic topological orders (without symmetry). We find explicit realizations for all the fermionic topological orders in the table. For example, we find that, up to invertible $p+\mathrm{i}p$ fermionic topological orders, there are only four fermionic topological orders with one non-trivial topological excitation: (1) the $K=\left(\begin{array}{cc}-1& 0\\ 0& 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order $2_{14/5}^B$ stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, (4) a primitive fermionic topological order that has a chiral central charge $c=\frac{1}{4}$, whose only topological excitation has a non-abelian statistics with a spin $s=\frac{1}{4}$ and a quantum dimension $d=1+\sqrt{2}$. We also proposed a categorical way to classify 2+1D invertible fermionic topological orders using modular extensions.

23 pages, 8 tables