Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
Recently, it was realized that quantum states of matter can be classified as long-range entangled states (i.e., with nontrivial topological order) and short-range entangled states (i.e., with trivial topological order). We can use group cohomology class H[superscript d](SG,R/Z) to systematically des...
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| Format: | Article |
| Language: | en_US |
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American Physical Society
2014
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| Online Access: | http://hdl.handle.net/1721.1/88947 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1811072336834592768 |
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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 | Recently, it was realized that quantum states of matter can be classified as long-range entangled states (i.e., with nontrivial topological order) and short-range entangled states (i.e., with trivial topological order). We can use group cohomology class H[superscript d](SG,R/Z) to systematically describe the SRE states with a symmetry SG [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in d-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e., to measure the elements in H[superscript d](G,R/Z) that label different SPT states. For example, 2+1-dimensional bosonic SPT states with Z[subscript n] symmetry are classified by a Z[subscript n] integer m ∈ H[superscript 3](Z[subscript n],R/Z) = Z[subscript n]. We find that n identical monodromy defects, in a Z[subscript n] SPT state labeled by m, carry a total Z[subscript n] charge 2m (which is not a multiple of n in general). |
| first_indexed | 2024-09-23T09:04:23Z |
| format | Article |
| id | mit-1721.1/88947 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:04:23Z |
| publishDate | 2014 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/889472022-09-30T13:14:13Z Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang Recently, it was realized that quantum states of matter can be classified as long-range entangled states (i.e., with nontrivial topological order) and short-range entangled states (i.e., with trivial topological order). We can use group cohomology class H[superscript d](SG,R/Z) to systematically describe the SRE states with a symmetry SG [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in d-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e., to measure the elements in H[superscript d](G,R/Z) that label different SPT states. For example, 2+1-dimensional bosonic SPT states with Z[subscript n] symmetry are classified by a Z[subscript n] integer m ∈ H[superscript 3](Z[subscript n],R/Z) = Z[subscript n]. We find that n identical monodromy defects, in a Z[subscript n] SPT state labeled by m, carry a total Z[subscript n] charge 2m (which is not a multiple of n in general). National Science Foundation (U.S.) (Grant DMR-1005541) 2014-08-21T15:00:30Z 2014-08-21T15:00:30Z 2014-01 2013-11 Article http://purl.org/eprint/type/JournalArticle 1098-0121 1550-235X http://hdl.handle.net/1721.1/88947 Wen, Xiao-Gang. “Symmetry-Protected Topological Invariants of Symmetry-Protected Topological Phases of Interacting Bosons and Fermions.” Phys. Rev. B 89, no. 3 (January 2014). © 2014 American Physical Society https://orcid.org/0000-0002-5874-581X en_US http://dx.doi.org/10.1103/PhysRevB.89.035147 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 American Physical Society |
| spellingShingle | Wen, Xiao-Gang Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions |
| title | Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions |
| title_full | Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions |
| title_fullStr | Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions |
| title_full_unstemmed | Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions |
| title_short | Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions |
| title_sort | symmetry protected topological invariants of symmetry protected topological phases of interacting bosons and fermions |
| url | http://hdl.handle.net/1721.1/88947 https://orcid.org/0000-0002-5874-581X |
| work_keys_str_mv | AT wenxiaogang symmetryprotectedtopologicalinvariantsofsymmetryprotectedtopologicalphasesofinteractingbosonsandfermions |