Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries

Tian Lan, Liang Kong, Xiao-Gang Wen
2017 Physical review B  
Gapped quantum liquids (GQL) 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 GQL for both bosonic and fermionic systems: 2+1D bosonic/fermionic GQLs with finite on-site symmetry are classified by non-degenerate unitary braided fusion categories over a symmetric fusion category (SFC) E, abbreviated as UMTC_/ E, together with their modular
more » ... th their modular extensions and total chiral central charges. The SFC E is Rep(G) for bosonic symmetry G, or sRep(G^f) for fermionic symmetry G^f. As a special case of the above result, we find that the modular extensions of Rep(G) classify the 2+1D bosonic SPT states of symmetry G, while the c=0 modular extensions of sRep(G^f) 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 it is not clear if free-fermion constructions can 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× 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.
doi:10.1103/physrevb.95.235140 fatcat:sso45oubgvg6dovowgnvsz4khy