In 2+1-dimensional space-time, gapped quantum states are always gapped quantum liquids (GQL) which include both topologically ordered states (with long range entanglement) and symmetry protected topological (SPT) states (with short range entanglement). In this paper, we propose a classification of 2+1D GQLs for both bosonic and fermionic systems: 2+1D bosonic/fermionic GQLs with finite on-site symmetry are classified by nondegenerate unitary braided fusion categories over a symmetric fusion category (SFC) $\mathcal{E}$, abbreviated as ${\mathrm{UMTC}}_{/\mathcal{E}}$, together with their modular extensions and total chiral central charges. In our classification, SFC $\mathcal{E}$ describes the symmetry, which is $\text{Rep}\left(G\right)$ for bosonic symmetry $G$, or $\text{sRep}\left(G^f\right)$ for fermionic symmetry $G^f$. As a special case of the above result, we find that the modular extensions of $\text{Rep}\left(G\right)$ classify the 2+1D bosonic SPT states of symmetry $G$, while the $c=0$ modular extensions of $\text{sRep}\left(G^f\right)$ classify the 2+1D fermionic SPT states of symmetry $G^f$. Many fermionic SPT states are studied based on the constructions from free-fermion models. But free-fermion constructions cannot produce all fermionic SPT states. Our classification does not have such a drawback. We show that, for interacting 2+1D fermionic systems, there are exactly 16 superconducting phases with no symmetry and no fractional excitations (up to $E_8$ bosonic quantum Hall states). Also, there are exactly 8 $Z_2\times Z_2^f$-SPT phases, 2 $Z_8^f$-SPT phases, and so on. Besides, we show that two topological orders with identical bulk excitations and central charge always differ by the stacking of the SPT states of the same symmetry.

• Received 13 February 2017

DOI:https://doi.org/10.1103/PhysRevB.95.235140