Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries
We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category E to describe the symmetry. Here, E=sRep(Z[subscript 2][supers...
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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/1721.1/106620 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1811078074226180096 |
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| author | Lan, Tian Kong, Liang Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Lan, Tian Kong, Liang Wen, Xiao-Gang |
| author_sort | Lan, Tian |
| collection | MIT |
| description | We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category E to describe the symmetry. Here, E=sRep(Z[subscript 2][superscript f]) describing particles in a fermionic product state without symmetry, or E=sRep(G[superscript f]) [E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the K=([−1 over 0][0 over 2]) fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=1/4, whose only topological excitation has non-Abelian statistics with spin s=1/4 and quantum dimension d=1√2. |
| first_indexed | 2024-09-23T10:52:58Z |
| format | Article |
| id | mit-1721.1/106620 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T10:52:58Z |
| publishDate | 2017 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/1066202022-09-30T23:41:28Z Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries Lan, Tian Kong, Liang Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category E to describe the symmetry. Here, E=sRep(Z[subscript 2][superscript f]) describing particles in a fermionic product state without symmetry, or E=sRep(G[superscript f]) [E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the K=([−1 over 0][0 over 2]) fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=1/4, whose only topological excitation has non-Abelian statistics with spin s=1/4 and quantum dimension d=1√2. National Science Foundation (U.S.) (Grant DMR-1005541) National Natural Science Foundation (China) (Grant 11274192) Templeton Foundation (Grant 39901) 2017-01-25T19:18:14Z 2017-01-25T19:18:14Z 2016-10 2016-07 2016-10-10T22:00:19Z Article http://purl.org/eprint/type/JournalArticle 2469-9950 2469-9969 http://hdl.handle.net/1721.1/106620 Lan, Tian, Liang Kong, and Xiao-Gang Wen. “Theory of (2+1)-Dimensional Fermionic Topological Orders and Fermionic/bosonic Topological Orders with Symmetries.” Physical Review B 94.15 (2016): n. pag. © 2016 American Physical Society https://orcid.org/0000-0002-5874-581X en http://dx.doi.org/10.1103/PhysRevB.94.155113 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 | Lan, Tian Kong, Liang Wen, Xiao-Gang Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries |
| title | Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries |
| title_full | Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries |
| title_fullStr | Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries |
| title_full_unstemmed | Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries |
| title_short | Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries |
| title_sort | theory of 2 1 dimensional fermionic topological orders and fermionic bosonic topological orders with symmetries |
| url | http://hdl.handle.net/1721.1/106620 https://orcid.org/0000-0002-5874-581X |
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