Hierarchy Construction and Non-Abelian Families of Generic Topological Orders
We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We ref...
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| Format: | Article |
| Language: | English |
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American Physical Society
2017
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| Online Access: | http://hdl.handle.net/1721.1/111597 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1826208082996232192 |
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| author | Lan, Tian Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Lan, Tian Wen, Xiao-Gang |
| author_sort | Lan, Tian |
| collection | MIT |
| description | We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalence class (the orbit of the hierarchy construction) as “the non-Abelian family.” Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups. |
| first_indexed | 2024-09-23T14:00:15Z |
| format | Article |
| id | mit-1721.1/111597 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:00:15Z |
| publishDate | 2017 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/1115972022-10-01T18:30:49Z Hierarchy Construction and Non-Abelian Families of Generic Topological Orders Lan, Tian Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalence class (the orbit of the hierarchy construction) as “the non-Abelian family.” Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups. National Science Foundation (U.S.) (Grant DMR-1506475) 2017-09-18T18:13:24Z 2017-09-18T18:13:24Z 2017-07 2017-05 2017-07-27T22:00:04Z Article http://purl.org/eprint/type/JournalArticle 0031-9007 1079-7114 http://hdl.handle.net/1721.1/111597 Lan, Tian and Wen, Xiao-Gang. "Hierarchy Construction and Non-Abelian Families of Generic Topological Orders." Physical Review Letters 119, 4: 040403 © 2017 American Physical Society https://orcid.org/0000-0002-5874-581X en http://dx.doi.org/10.1103/PhysRevLett.119.040403 Physical Review Letters Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. American Physical Society application/pdf American Physical Society American Physical Society |
| spellingShingle | Lan, Tian Wen, Xiao-Gang Hierarchy Construction and Non-Abelian Families of Generic Topological Orders |
| title | Hierarchy Construction and Non-Abelian Families of Generic Topological Orders |
| title_full | Hierarchy Construction and Non-Abelian Families of Generic Topological Orders |
| title_fullStr | Hierarchy Construction and Non-Abelian Families of Generic Topological Orders |
| title_full_unstemmed | Hierarchy Construction and Non-Abelian Families of Generic Topological Orders |
| title_short | Hierarchy Construction and Non-Abelian Families of Generic Topological Orders |
| title_sort | hierarchy construction and non abelian families of generic topological orders |
| url | http://hdl.handle.net/1721.1/111597 https://orcid.org/0000-0002-5874-581X |
| work_keys_str_mv | AT lantian hierarchyconstructionandnonabelianfamiliesofgenerictopologicalorders AT wenxiaogang hierarchyconstructionandnonabelianfamiliesofgenerictopologicalorders |