Classification of two-dimensional fermionic and bosonic topological orders
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)-dimensional (2+1)D...
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| Format: | Article |
| Language: | English |
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American Physical Society
2015
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| Online Access: | http://hdl.handle.net/1721.1/96372 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1811097047457071104 |
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| author | Gu, Zheng-Cheng Wang, Zhenghan Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Gu, Zheng-Cheng Wang, Zhenghan Wen, Xiao-Gang |
| author_sort | Gu, Zheng-Cheng |
| collection | MIT |
| description | 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)-dimensional (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 fermionic topological orders with gappable edge in terms of the following set of data (N[ij over k], F[ij over k], F[ijm,αβ over jkn,χδ], d[subscript 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 F[ij over k] = 0, our result recovers the previous classification of 2 + 1D bosonic topological orders with gappable edge. |
| first_indexed | 2024-09-23T16:53:29Z |
| format | Article |
| id | mit-1721.1/96372 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:53:29Z |
| publishDate | 2015 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/963722022-10-03T08:57:06Z Classification of two-dimensional fermionic and bosonic topological orders Gu, Zheng-Cheng Wang, Zhenghan Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang 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)-dimensional (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 fermionic topological orders with gappable edge in terms of the following set of data (N[ij over k], F[ij over k], F[ijm,αβ over jkn,χδ], d[subscript 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 F[ij over k] = 0, our result recovers the previous classification of 2 + 1D bosonic topological orders with gappable edge. National Science Foundation (U.S.) (Grant DMR-1005541) National Natural Science Foundation (China) (Grant 11274192) BMO Financial Group Templeton Foundation (Grant 39901) Canada. Industry Canada Ontario. Ministry of Research and Innovation 2015-04-03T16:30:43Z 2015-04-03T16:30:43Z 2015-03 2015-01 2015-03-31T22:00:06Z Article http://purl.org/eprint/type/JournalArticle 1098-0121 1550-235X http://hdl.handle.net/1721.1/96372 Gu, Zheng-Cheng, Zhenghan Wang, and Xiao-Gang Wen. "Classification of two-dimensional fermionic and bosonic topological orders." Phys. Rev. B 91, 125149 (March 2015). © 2015 American Physical Society https://orcid.org/0000-0002-5874-581X en http://dx.doi.org/10.1103/PhysRevB.91.125149 Physical Review B 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 | Gu, Zheng-Cheng Wang, Zhenghan Wen, Xiao-Gang Classification of two-dimensional fermionic and bosonic topological orders |
| title | Classification of two-dimensional fermionic and bosonic topological orders |
| title_full | Classification of two-dimensional fermionic and bosonic topological orders |
| title_fullStr | Classification of two-dimensional fermionic and bosonic topological orders |
| title_full_unstemmed | Classification of two-dimensional fermionic and bosonic topological orders |
| title_short | Classification of two-dimensional fermionic and bosonic topological orders |
| title_sort | classification of two dimensional fermionic and bosonic topological orders |
| url | http://hdl.handle.net/1721.1/96372 https://orcid.org/0000-0002-5874-581X |
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