Gapless edges of 2d topological orders and enriched monoidal categories

Kong, Liang; Zheng, Hao

doi:10.1016/j.nuclphysb.2017.12.007

Cited by 24 publications

(43 citation statements)

References 91 publications

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“…In particular, we will work out explicitly which boundary-bulk CFT is produced by this dimensional reduction process. It turns out that all boundary-bulk RCFT's can be obtained in this way (first announced in [KZ3]).…”

Section: Dimensional Reduction To Boundary-bulk Cft'smentioning

confidence: 97%

A mathematical theory of gapless edges of 2d topological orders. Part I

Kong

Zheng

2020

J. High Energ. Phys.

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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. Mathematically, this relation is stated precisely 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 explain how to gap out non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundarybulk 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.

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Section: Dimensional Reduction To Boundary-bulk Cft'smentioning

confidence: 97%

A mathematical theory of gapless edges of 2d topological orders. Part I

Kong

Zheng

2020

J. High Energ. Phys.

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“…In last section, we pointed out the boundary of 2+1D Z 2 topological order (which has a non-invertible gravitational anomaly) has a UV completion described by a lattice model (32), with a constraint on the Hilbert space i σ z i = 1. It is the constraint i σ z i = 1 that makes the Hilbert space non-local.…”

Section: Non-invertible Gravitational Anomaly and "Non-locality" Omentioning

confidence: 99%

“…To obtain the partition function of the anomalous CFT, let us first consider the partition function of the transverse Ising model (32) at critical point U = J. There are four partition functions Z ax,at for the transverse Ising model, with different Z 2 boundary conditions a x = ±1 and a t = ±1.…”

Section: B a Gapless Boundary Of The 2+1d Z2 Topological Ordermentioning

confidence: 99%

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Noninvertible anomalies and mapping-class-group transformation of anomalous partition functions

Wen

2019

Phys. Rev. Research

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Recently, it was realized that anomalies can be completely classified by topological orders, symmetry protected topological orders, and symmetry enriched topological orders in one higher dimension. The anomalies that people used to study are invertible anomalies that correspond to invertible topological orders and/or symmetry protected topological orders in one higher dimension. In this paper, we introduce a notion of non-invertible anomaly, which describes the boundary of generic topological order. A key feature of non-invertible anomaly is that it has several partition functions. Under the mapping class group transformation of space-time, those partition functions transform in a certain way characterized by the data of the corresponding topological order in one higher dimension. In fact, the anomalous partition functions transform in the same way as the degenerate ground states of the corresponding topological order in one higher dimension. This general theory of non-invertible anomaly may have wide applications. As an example, we show that the irreducible gapless boundary of 2+1D double-semion topological order must have central charge c = c ≥ 25 28 . CONTENTS

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“…The above defects are expected and inevitable because the boundary-bulk correspondence is many-to-one, which already happens in the chiral cases [45][46][47] . The same bulk theory can share many different boundary theories, even gapless ones 48 . Consequently, one cannot extract boundary data purely from bulk information.…”

Section: A General Boundary Conditionsmentioning

confidence: 99%

Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases

Luo

Pankovich

et al. 2019

Phys. Rev. B

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We study the correspondence between boundary excitation distribution spectrum of non-chiral topological orders on an open surface M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as M. Specifically, we focus on the case of generalized string-net models and discuss the cases where M is a disk or a cylinder. When M has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, ZN theories, and the simple non-abelian case of doubled Fibonacci are demonstrated. arXiv:1806.07794v3 [cond-mat.str-el] 12 May 2019where the decomposition matrix M is 4 × 2-dimensional, with the first subscript taking values from {1, m, e, }, i.e. the output category, and the second subscript from {0, 1}, the input category. 59 Specifically,

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Gapless edges of 2d topological orders and enriched monoidal categories

Cited by 24 publications

References 91 publications

A mathematical theory of gapless edges of 2d topological orders. Part I

A mathematical theory of gapless edges of 2d topological orders. Part I

Noninvertible anomalies and mapping-class-group transformation of anomalous partition functions

Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases

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