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arXiv 1507.04673cond-mat.str-el
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具有对称性的2+ 1维费米子拓扑序和费米子/玻色子拓扑序理论

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

Tian Lan, Liang Kong, Xiao-Gang Wen

备注：
23 pages, 8 tables
发表信息：
Phys. Rev. B 94, 155113 (2016)
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摘要：

我们提出，直到可逆拓扑序，不具有对称性的2+1D费米子拓扑序和具有对称性的2+ 1D费米子/玻色子拓扑序$G$都可以用对称融合范畴（SFC）上的非退化酉编织融合范畴（UBFC）来分类; SFC描述了不具有对称性的费米子产物态或具有对称性的费米子/玻色子产物态$G$，并且UBFC具有模块化扩展。基于融合系数$N^{ij}_k$和自旋$s_i$，我们发展了一个简化的SFC上非简并UBFC理论。这使得我们可以得到一个包含所有2+1D费米子拓扑序（没有对称性）的列表。我们在表中找到了所有费米拓扑序的明确实现。例如，我们发现，直到可逆的$p+\hspace{1pt}\mathrm{i}\hspace{1pt} p$费米拓扑序，只有四个费米拓扑序具有一个非平凡的拓扑激发：（1）$K={\scriptsize \begin{pmatrix} -1&0\\0&2\end{pmatrix}}$分数量子霍尔态，（2）与费米子乘积态叠加的斐波那契玻色子拓扑序$2^B_{14/5}$，（3）前一个的时间反演共轭，（4）具有手征中心荷$c=\frac14$的原始费米子拓扑序，其唯一的拓扑激发具有自旋$s=\frac14$和量子维度$d=1+\sqrt{2}$的非阿贝尔统计。我们还提出了一种利用模扩张对2+ 1维可逆费米子拓扑序进行分类的方法。

英文摘要：

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 $N^{ij}_k$ 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 $p+\hspace{1pt}\mathrm{i}\hspace{1pt} p$ fermionic topological orders, there are only four fermionic topological orders with one non-trivial topological excitation: (1) the $K={\scriptsize \begin{pmatrix} -1&0\\0&2\end{pmatrix}}$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order $2^B_{14/5}$ 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 $c=\frac14$, whose only topological excitation has a non-abelian statistics with a spin $s=\frac14$ 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.

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