A classification of 2D fermionic and bosonic topological orders

Details A classification of 2D fermionic and bosonic topological orders A classification of 2D fermionic and bosonic topological orders

2010-10-07

arxiv.org/abs/1010.1517v1

Physics

Condensed Matter

Strongly Correlated Electrons

34 page 15 figures, RevTeX4+tikz

Scientific paper

The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to systematically describe topological orders in 2D boson systems. The two approaches reveal that the mathematical framework for topological order is closely related to (projective) tensor category theory. In this paper, we generalize those systematic descriptions of topological orders to 2D fermion systems. We find that a large class of topological orders in 2D fermion systems can be systematically characterized by the following set of data $(N_{ijk}, N^f_{ijk}, F^{ijm,\al\bt,a}_{jkn,\chi\del,b}, O_{i,a}^{jk,\al\bt})$ 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. Since topological orders in 2D boson systems described by the string-net approach have a deep relationship with the Turaev-Viro states summation invariance of 3-manifolds, it is natural to expect our classification of topological orders in 2D fermionic systems will generalize the famous Turaev-Viro invariance to its fermionic version. When $N^f_{ijk}=0$, our results provide a classification of bosonic topological orders that generalizes the string-net classification.

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