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

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          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 \(N^{ij}_k\) 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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          Author and article information

          Journal
          2015-07-16
          2015-08-27
          Article
          1507.04673
          4bbbf1af-b88f-4452-833e-d7d2d802e1fc

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          23 pages, 8 tables
          cond-mat.str-el

          Condensed matter
          Condensed matter

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