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arXiv:1911.08470 · Physical Review Research

Gapped domain walls between 2+1D topologically ordered states

Tian Lan, Xueda Wen, Liang Kong, Xiao-Gang Wen

Published Jun 12, 2020• 17 citations
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Abstract

This work proposed a systematic theory about the gapped domain walls between 2+1D topological ordered states. The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. Gapped walls are characterized by the wave function overlaps, or fixed-point partition functions, which must satisfy conditions given by the mapping-class-group representations. By solving these conditions, one can determine all gapped domain walls between given topological ordered states.

DOI
10.1103/physrevresearch.2.023331

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This work proposed a systematic theory about the gapped domain walls between 2+1D topological ordered states. The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. Gapped walls are characterized by the wave function overlaps, or fixed-point partition functions, which must satisfy conditions given by the mapping-class-group representations. By solving these conditions, one can determine all gapped domain walls between given topological ordered states.
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.
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.
We propose a way-universal wave-function overlap-to extract universal topological data from generic ground states of gapped systems in any dimensions. Those extracted topological data might fully characterize the topological orders with a gapped or gapless boundary. For nonchiral topological orders in (2+1)D, these universal topological data consist of two matrices S and T, which generate a projective representation of SL(2,Z) on the degenerate ground state Hilbert space on a torus. For topological orders with a gapped boundary in higher dimensions, these data constitute a projective representation of the mapping class group MCG(M^{d}) of closed spatial manifold M^{d}. For a set of simple models and perturbations in two dimensions, we show that these quantities are protected to all orders in perturbation theory. These overlaps provide a much more powerful alternative to the topological entanglement entropy and allow for more efficient numerical implementations.
We introduce a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible G×Rep(G)×GG\times \text{Rep}(G)\times GG×Rep(G)×G symmetry and invertible domain walls in the quantum double associated with the group GGG. We prove this correspondence when GGG is Abelian and provide evidence for the general case by studying the quantum double model for G=S3G=S_3G=S3​. We also use our method to construct \emph{anchoring domain walls}, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.
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.