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arXiv:1408.6514 · Physical Review Letters

Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy

Tian Lan, Juven Wang, Xiao-Gang Wen

Published Feb 18, 2015• 155 citations
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Abstract

Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

DOI
10.1103/physrevlett.114.076402

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Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.
The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize the gapped domain walls (or the boundaries). The mapping-class-group representations give rise to conditions that must be satisfied by the fixed-point partition functions, which leads to a systematic theory. Such conditions can be viewed as bulk topological order determining the (non-invertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (non-invertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.
We introduce the concept of boundary degeneracy, as the ground state degeneracy of topologically ordered states on a compact orientable spatial manifold with gapped boundaries. We emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the Z2{Z}_{2}Z2​ toric code and Z2{Z}_{2}Z2​ double-semion model [more generally, the Zk{Z}_{k}Zk​ gauge theory and the $U{(1)}_{k}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}k}$ nonchiral fractional quantum Hall state at even integer kkk] can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.
This is the second part of a two-part work on the unified mathematical theory of gapped and gapless edges of 2+1D topological orders. In Part I, we have developed the mathematical theory of chiral gapless edges. In Part II, we study boundary-bulk relation and non-chiral gapless edges. In particular, we explain how the notion of the center of an enriched monoidal category naturally emerges from the boundary-bulk relation. After the study of 0+1D gapless walls, we give the complete boundary-bulk relation for 2+1D topological orders with chiral gapless edges (including gapped edges) and 0d walls between edges. This relation is stated precisely and proved rigorously as a monoidal equivalence, which generalizes the functoriality of the usual Drinfeld center to an enriched setting. We also develop the mathematical theory of non-chiral gapless edges and 0+1D walls, and explain how to gap out certain non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundary-bulk rational CFT's can be recovered from 2d topological orders with chiral gapless edges via a dimensional reduction process. This provides physical meanings to some mysterious connections between mathematical results in fusion categories and those in rational CFT's.
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).
Domain walls between different topological phases are one of the most interesting phenomena that reveal the nontrivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group GGG. As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as S3{S}_{3}S3​) in 2D2\mathrm{D}2D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.
The structure of extrinsic defects in topologically ordered states of matter is host to a rich set of universal physics. Extrinsic defects in 2+12+12+1-dimensional topological states include linelike defects, such as boundaries between topologically distinct states, and pointlike defects, such as junctions between different line defects. Gapped boundaries in particular can themselves be topologically distinct, and the junctions between them can localize topologically protected zero modes, giving rise to topological ground-state degeneracies and projective non-Abelian statistics. In this paper, we develop a general theory of point defects and gapped line defects in 2+12+12+1-dimensional Abelian topological states. We derive a classification of topologically distinct gapped boundaries in terms of certain maximal subgroups of quasiparticles with mutually bosonic statistics, called Lagrangian subgroups. The junctions between different gapped boundaries provide a general classification of point defects in topological states, including as a special case the twist defects considered in previous works. We derive a general formula for the quantum dimension of these point defects and a general understanding of their localized ``parafermion'' zero modes and we define a notion of projective non-Abelian statistics for them. The critical phenomena between topologically distinct gapped boundaries can be understood in terms of a general class of quantum spin chains or, equivalently, ``generalized parafermion'' chains. This provides a way of realizing exotic 1+1D1+1\mathrm{D}1+1D generalized parafermion conformal field theories in condensed-matter systems.