Topological orders and factorization homology

Abstract

In the study of 2d (the space dimension) topological orders, it is well-known that bulk excitations are classified by unitary modular tensor categories. But these categories only describe the local observables on an open 2-disk in the long wave length limit. For example, the notion of braiding only makes sense locally. It is natural to ask how to obtain global observables on a closed surface. The answer is provided by the theory of factorization homology. We compute the factorization homology of a closed surface Σ with the coefficient given by a unitary modular tensor category, and show that the result is given by a pair (H, u Σ ), where H is the category of finite-dimensional Hilbert spaces and u Σ ∈ H is a distinguished object that coincides precisely with the Hilbert space assigned to the surface Σ in Reshetikhin-Turaev TQFT. We also generalize this result to a closed stratified surface decorated by anomaly-free topological defects of codimension 0,1,2. This amounts to compute the factorization homology of a stratified surface with a coefficient system satisfying an anomaly-free condition.The study of topological orders has attracted a lot of attentions in recent years. In this work, we show how to compute the global observables of an anomaly-free 2d topological order on a closed 2d manifold. Here 2d is the space dimension. We also generalize the computation to closed stratified 2d manifolds decorated by anomaly-free topological defects.By an anomaly-free 2d topological order, we mean a 2d topological order that can be realized by 2d lattice models [KW]. It is known that anomaly-free 2d topological orders are classified by unitary modular tensor categories (UMTC's) (up to E 8 quantum Hall states which will be ignored completely in this work). The objects in the UMTC correspond to topological excitations, which are all particle-like (i.e. 0d) topological defects (also called anyons) and completely local [Ki, LW, KK]. These topological excitations can be moved (by string operators), fused and braided. These fusion-braiding structures are precisely given by the data and axioms of a UMTC. The trivial 2d topological order is given by the simplest UMTC: the category of finitedimensional Hilbert spaces, denoted by H.However, what has not been clarified nor emphasized in physics literature is that the fusionbraiding structures of topological excitations are only local observables defined in an open 2-disk. For example, the double braiding of two objects x and y in a UMTC, loosely speaking, corresponds to moving the particle-like topological excitation x around another y along a circular path. This double braiding only makes sense locally, for example, in an open 2-disk. If x and y are located on a sphere, a circular path around the topological excitation y is contractible. Therefore, the double braiding does not make sense on a sphere at all. It means that the braiding structure is not a global observable. Then an obvious question is what the global observables are. The answer to this question is provided by the theor...

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

“…If one inserts a finite number of particlelike excitations x 1 , · · · ,x r on the closed surface, one simply obtain the Hilbert space hom C (1,x 1 ⊗ · · · ⊗ x r ), which is also the space of degenerate ground states. This result remains to be true for all closed two-dimensional manifolds with topological gapped defects and with two cells decorated by different phases [53]. This includes the cases that the topological order is defined on any surfaces with boundaries.…”

“…This suggests that the category C should equipped with a natural functor to the category of finite dimensional Hilbert spaces, which is a factorization homology M of a UMTC M [53]. So we expect that we should be able to embed C into a UMTC M such that the embedding naturally descends to a functor C → M on factorization homologies.…”

“…[53] that factorization homology of a UMTC C over a closed two-dimensional manifold is given by the category of finite dimensional Hilbert spaces. If one inserts a finite number of particlelike excitations x 1 , · · · ,x r on the closed surface, one simply obtain the Hilbert space hom C (1,x 1 ⊗ · · · ⊗ x r ), which is also the space of degenerate ground states.…”

Gapless edges of 2d topological orders and enriched monoidal categories

“…Similar to the fusion of gapped domain walls [KK,FS2,LWW,HW,AKZ,Ka], we can obtain a new gapless edge of the 2d bulk phase (C, c) by fusing a canonical edge (V, B ) of (B, c) with a gapped wall M between (B, c) and (C, c). We denote the new gapless edge obtained from this fusion by (V, B ) (B,c) M, or graphically as follows:…”

Topological insulators and the Kane–Mele invariant: Obstruction and localization theory

“…We write Aut qu (PH) for the automorphisms of PH which leave transition probabilities invariant, and Aut qu (H) for the group of unitary and antiunitary operators on H. By a famous theorem of Wigner [78], there is a short exact sequence If a quantum system comes endowed with a specified Hamiltonian H on H, we would like to restrict to the subgroup of symmetries Aut qu (H, H) ⊂ Aut qu (H) which projectively commute with the Hamiltonian. That is, we demand that there is a map c : Aut qu (H, H) × H → U (1) such that T Hψ = c(T, ψ) H T ψ for all ψ ∈ H and T ∈ Aut qu (H, H). By refining the treatment in [27], we show in Section 3.3 directly from this setup that the only consistent choices of such maps are given by continuous Z 2 -gradings c : Aut qu (H, H) → Z 2 .…”