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arXiv:1311.1784 · Physical Review B

Topological quasiparticles and the holographic bulk-edge relation in (2+1) -dimensional string-net models

Tian Lan, Xiao-Gang Wen

Published Sep 9, 2014• 123 citations
arXivHTMLPDF

Abstract

String-net models allow us to systematically construct and classify (2+1)-dimensional [(2+1)D] topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wave function, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method, Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wave function (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory [i.e., twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$]. Our result can also be viewed from an angle of holographic bulk-boundary relation. The (1+1)-dimensional [(1+1)D] anomalous topological orders, that can appear as edges of (2+1)D topological states, are classified by UFCs which describe the fusion of quasiparticles in (1+1)D. The (1+1)D anomalous edge topological order uniquely determines the (2+1)D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).

DOI
10.1103/physrevb.90.115119

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String-net models allow us to systematically construct and classify (2+1)-dimensional [(2+1)D] topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wave function, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method, Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wave function (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory [i.e., twisted quantum double D\ensuremathα(G){D}^{\ensuremath{\alpha}}(G)D\ensuremathα(G)]. Our result can also be viewed from an angle of holographic bulk-boundary relation. The (1+1)-dimensional [(1+1)D] anomalous topological orders, that can appear as edges of (2+1)D topological states, are classified by UFCs which describe the fusion of quasiparticles in (1+1)D. The (1+1)D anomalous edge topological order uniquely determines the (2+1)D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).
Topological orders are a class of phases of matter that beyond the Landau symmetry breaking paradigm. The two (spatial) dimensional (2d) topological orders have been thoroughly studied. It is known that they can be fully classified by a unitary modular tensor category (UMTC) and a chiral central charge c. And a class of 2d topological orders whose boundary are gappable can be systematically constructed by Levin-Wen model whose ground states are string-net condensed states. Previously, the three spatial dimensional topological orders have been classified based on their canonical boundary described by some special unitary fusion 2-category, 2VecGω2\mathcal{V}ec_G^ω2VecGω​ or an EF 2-category. However, a lattice realization of a three spatial dimensional topological orders with both canonical boundary and arbitrary boundaries are still lacking. In this paper, we construct a 3d membrane-net model based on spherical fusion 2-category, which can be used to systematically study all general 3d topological order with gapped boundary. The partition function and lattice Hamiltonian of the membrane-net model is constructed based on state sum of spherical fusion 2-category. We also construct the 3d tube algebra of the membrane-net model to study excitations in the model. We conjecture all intrinsic excitations in membrane-net model have a one-to-one correspondence with the irreducible central idempotents (ICI) of the 3d tube algebra. We also provide a universal framework to study mutual statistics of all excitations in 3d topological order through 3d tube algebra. Our approach can be straightforwardly generalized to arbitrary dimension.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.
Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. (4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. (5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a Z\mathbb{Z}Z class in 2+1D (with gapless edge), a Z2\mathbb{Z}_2Z2​ class in 4+1D (with gappable boundary), and a Z⊕Z\mathbb{Z}\oplus \mathbb{Z}Z⊕Z class in 6+1D (with gapless boundary).
The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in two-dimensional (2D) bosonic systems. The two approaches reveal that the mathematical framework for (2+1)2+1)2+1)-dimensional (2+1)D(2+1)\mathrm{D}(2+1)D bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of (2+1)D(2+1)\mathrm{D}(2+1)D fermionic topological orders with gappable edge in terms of the following set of data $({N}_{k}^{ij},{F}_{k}^{ij},{F}_{jkn,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{d}_{i})$, which satisfy a set of nonlinear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When Fkij=0{F}_{k}^{ij}=0Fkij​=0, our result recovers the previous classification of 2+1D2+1\mathrm{D}2+1D bosonic topological orders with gappable edge.
Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground-state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group MCG(M)\mathtt{MCG}(\mathcal{M})MCG(M) of the space manifold M\mathcal{M}M on which the systems live. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to the fusion algebra and statistics of particles and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contain information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular SSS and TTT matrices to any dimension.
In this work, we give a mathematical description of a chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk rational conformal field theories naturally emerge on certain gapless walls between two trivial phases.