Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

摘要

We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category £ to describe the symmetry. Here, ε= sRep(Z_2~f) describing particles in a fermionic product state without symmetry, or ε = sRep(G~f) [ε= Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry £ are classified by nondegenerate unitary braided fusion categories over £, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p + i p fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation:(1)the K= (-1 0 0 2) fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c = 1/4, whose only topological excitation has non-Abelian statistics with spin s = 1/4 and quantum dimension d = 1+2~(1/2).