arXiv:1602.05946 · Physical review. B./Physical review. B
Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries
Large classes of matter have been systematized before by using a pair of groups to classify all spontaneous symmetry breaking phases or by using a pair of groups plus projective representations to classify all gapped quantum phases of one-dimensional boson and fermion systems with various symmetries. In this paper, the authors go beyond this and use a pair of groups plus three braided fusion categories and a chiral central charge in order to classify gapped quantum phases of two-dimensional boson and fermion systems with various finite symmetries. This allows, among other things, to generate many group tables of two-dimensional gapped phases.
Large classes of matter have been systematized before by using a pair of groups to classify all spontaneous symmetry breaking phases or by using a pair of groups plus projective representations to classify all gapped quantum phases of one-dimensional boson and fermion systems with various symmetries. In this paper, the authors go beyond this and use a pair of groups plus three braided fusion categories and a chiral central charge in order to classify gapped quantum phases of two-dimensional boson and fermion systems with various finite symmetries. This allows, among other things, to generate many group tables of two-dimensional gapped phases.
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 E to describe the symmetry. Here, E=sRep(Z2f) describing particles in a fermionic product state without symmetry, or E=sRep(Gf)[E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, 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+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ 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=41, whose only topological excitation has non-Abelian statistics with spin s=41 and quantum dimension d=1+2.
This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories E⊂C⊂M plus the chiral central charge c. Here E is a symmetric fusion category, E=Rep(G) for boson systems or E=sRep(Gf) for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; C is the category of all the quasiparticle excitations in the bulk, containing E as its Müger center; M is a minimal modular extension of C, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.
We examine the interplay of symmetry and topological order in 2+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2f on the physical fermions is essentially the central extension of G by the Z2f fermion parity conservation that yields the fermionic symmetry Gf. We develop an algebraic theory of Gf symmetry defects for fermionic topological phases using G-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+1 dimensions, the quantum phases in the presence of a global symmetry group SG, and with topological order described by a finite gauge group GG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group G and its group 4-cocycle ω4∈H4[G;U(1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
Abstract In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and CFT-type gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial. We also study the categorical descriptions of the boundaries of these models. In the end, we obtain a classification of and the categorical descriptions of all 1-dimensional (spatial dimension) gapped quantum phases with a bosonic/fermionic finite onsite symmetry.