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

Abstract

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) E, abbreviated as UMTC /E , together with their modular extensions and total chiral central charges. In our classification, SFC E describes the symmetry, which 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 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 × Z f 2 -SPT phases, 2 Z f 8 -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.

Symmetry enriched U(1) quantum spin liquids

“…VI may be potentially helpful. Another possibly helpful formal approach is to generalize the categorical theory that is used to study 2d SETs to U(1) quantum spin liquids [9,11]. This may be possible because in both cases the excitations are all particle like, although there are infinitely many types of fractional excitations in a U(1) quantum spin liquid.…”

“…Thus symmetry protected distinctions between different SET phases may be much more striking than in topological band insulators. Though much of the early work on spin liquids dealt with SET phases, it is only in the last few years that there has been tremendous and systematic progress in understanding their full structure and classification in two-dimensional systems [1][2][3][4][5][6][7][8][9][10][11]. Some limited progress has been made for three-dimensional SET phases as well [12][13][14].…”

Matrix formulation for non-Abelian families

“…There is then a natural non-degenerate symmetric bilinear form K on Z κ induced from K, via K(f, g) = K(K(f ),K(g)) = f (K(g)) = g(K(f )). (19) If one chooses a basis of Z κ and the corresponding dual basis of Z κ , the matrix of K and K are inverse to each other.…”

Hierarchy Construction and Non-Abelian Families of Generic Topological Orders

“…A classic example of such a classification result is the "periodic table" for chemical elements. As for the topological ordered [1,2] phases of matter, which has drawn more and more research interest recently, we are already able to create some tables for them [3][4][5][6][7][8], via the theory of (pre-)modular categories. However, efforts are needed to further understand the tables, for example, to reveal some "periodic" structures in the table.…”

“…Thus, classifying all topological orders is the same as classifying all root states, namely, all states such that their Abelian anyons have trivial mutual statistics. In other words, we can try to generate all possible topological orders by constructing all the root states, which can be obtained by starting with an Abelian group G, extending its representation category RepðGÞ or sRepðG f Þ to a modular category [7,8] while requiring all the extra anyons to be non-Abelian (which is referred to as a non-Abelian modular extension). This is a promising…”