Symmetry-protected topological phases in noninteracting fermion systems
Symmetry-protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For noninteracting fermion systems with time reversal (T̂), charge conjugation (Ĉ), and/or U(1) (N̂) symmetr...
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| Format: | Article |
| Language: | en_US |
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American Physical Society
2012
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| Online Access: | http://hdl.handle.net/1721.1/71285 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1811068717982810112 |
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| author | Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang |
| author_sort | Wen, Xiao-Gang |
| collection | MIT |
| description | Symmetry-protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For noninteracting fermion systems with time reversal (T̂), charge conjugation (Ĉ), and/or U(1) (N̂) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as T̂N̂T̂[superscript −1]=N̂ or T̂N̂T̂−1=−N̂. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free-fermion phases for those possible symmetry groups. In particular, we can view the U(1) as a spin rotation. We find that superconductors with the S[subscript z] spin-rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the Sz spin-rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free-fermion phases can be realized by electron systems with certain symmetries. We also point out that, to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (−)[superscript N̂]. The full symmetry group is actually a projective symmetry group. |
| first_indexed | 2024-09-23T08:00:04Z |
| format | Article |
| id | mit-1721.1/71285 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:00:04Z |
| publishDate | 2012 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/712852022-09-23T10:10:14Z Symmetry-protected topological phases in noninteracting fermion systems Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang Wen, Xiao-Gang Symmetry-protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For noninteracting fermion systems with time reversal (T̂), charge conjugation (Ĉ), and/or U(1) (N̂) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as T̂N̂T̂[superscript −1]=N̂ or T̂N̂T̂−1=−N̂. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free-fermion phases for those possible symmetry groups. In particular, we can view the U(1) as a spin rotation. We find that superconductors with the S[subscript z] spin-rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the Sz spin-rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free-fermion phases can be realized by electron systems with certain symmetries. We also point out that, to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (−)[superscript N̂]. The full symmetry group is actually a projective symmetry group. National Science Foundation (U.S.) (grant DMR-1005541) 2012-07-02T14:23:23Z 2012-07-02T14:23:23Z 2012-02 2011-12 Article http://purl.org/eprint/type/JournalArticle 1098-0121 1550-235X http://hdl.handle.net/1721.1/71285 Wen, Xiao-Gang. “Symmetry-protected Topological Phases in Noninteracting Fermion Systems.” Physical Review B 85.8 (2012). ©2012 American Physical Society https://orcid.org/0000-0002-5874-581X en_US http://dx.doi.org/10.1103/PhysRevB.85.085103 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. application/pdf American Physical Society APS |
| spellingShingle | Wen, Xiao-Gang Symmetry-protected topological phases in noninteracting fermion systems |
| title | Symmetry-protected topological phases in noninteracting fermion systems |
| title_full | Symmetry-protected topological phases in noninteracting fermion systems |
| title_fullStr | Symmetry-protected topological phases in noninteracting fermion systems |
| title_full_unstemmed | Symmetry-protected topological phases in noninteracting fermion systems |
| title_short | Symmetry-protected topological phases in noninteracting fermion systems |
| title_sort | symmetry protected topological phases in noninteracting fermion systems |
| url | http://hdl.handle.net/1721.1/71285 https://orcid.org/0000-0002-5874-581X |
| work_keys_str_mv | AT wenxiaogang symmetryprotectedtopologicalphasesinnoninteractingfermionsystems |