Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically desc...
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| Format: | Article |
| Language: | English |
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American Physical Society
2014
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| Online Access: | http://hdl.handle.net/1721.1/90379 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1811074211925458944 |
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| author | Gu, Zheng-Cheng Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Gu, Zheng-Cheng Wen, Xiao-Gang |
| author_sort | Gu, Zheng-Cheng |
| collection | MIT |
| description | Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T[superscript 2] = 1 time-reversal Z[T over 2] and fermion-number-parity Z[f over 2] symmetries described by a full symmetry group Z[T over 2] × Z[f over 2]). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z[subscript 2] × Z[f over 2]. Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken. |
| first_indexed | 2024-09-23T09:45:44Z |
| format | Article |
| id | mit-1721.1/90379 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:45:44Z |
| publishDate | 2014 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/903792022-09-30T16:38:12Z Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory Gu, Zheng-Cheng Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T[superscript 2] = 1 time-reversal Z[T over 2] and fermion-number-parity Z[f over 2] symmetries described by a full symmetry group Z[T over 2] × Z[f over 2]). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z[subscript 2] × Z[f over 2]. Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken. National Science Foundation (U.S.) (Grant DMR-1005541) Templeton Foundation 2014-09-26T12:43:23Z 2014-09-26T12:43:23Z 2014-09 2014-08 2014-09-23T22:00:16Z Article http://purl.org/eprint/type/JournalArticle 1098-0121 1550-235X http://hdl.handle.net/1721.1/90379 Gu, Zheng-Cheng, and Xiao-Gang Wen. "Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory." Phys. Rev. B 90, 115141 (September 2014). © 2014 American Physical Society https://orcid.org/0000-0002-5874-581X en http://dx.doi.org/10.1103/PhysRevB.90.115141 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 Wen, Xiao-Gang Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory |
| title | Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory |
| title_full | Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory |
| title_fullStr | Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory |
| title_full_unstemmed | Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory |
| title_short | Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory |
| title_sort | symmetry protected topological orders for interacting fermions fermionic topological nonlinear σ models and a special group supercohomology theory |
| url | http://hdl.handle.net/1721.1/90379 https://orcid.org/0000-0002-5874-581X |
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