{"title":"对称性丰富的拓扑量子自旋液体的分类","authors":"Weicheng Ye, Liujun Zou","doi":"10.1103/physrevx.14.021053","DOIUrl":null,"url":null,"abstract":"We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary symmetry, and may permute anyons. The framework applies to all types of lattices and can systematically distinguish different lattice systems with the same symmetry group using their quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">U</mi><mo stretchy=\"false\">(</mo><mn>1</mn><msub><mo stretchy=\"false\">)</mo><mrow><mn>2</mn><mi>N</mi></mrow></msub></math> chiral states and non-Abelian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msup><mrow><mi>Ising</mi></mrow><mrow><mo stretchy=\"false\">(</mo><mi>ν</mi><mo stretchy=\"false\">)</mo></mrow></msup></mrow></math> states enriched by a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>6</mn><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math> or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>4</mn><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math> symmetry and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mi mathvariant=\"double-struck\">Z</mi><mi>N</mi></msub></math> topological orders and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">U</mi><mo stretchy=\"false\">(</mo><mn>1</mn><msub><mo stretchy=\"false\">)</mo><mrow><mn>2</mn><mi>N</mi></mrow></msub><mo>×</mo><mi mathvariant=\"normal\">U</mi><mo stretchy=\"false\">(</mo><mn>1</mn><msub><mo stretchy=\"false\">)</mo><mrow><mo>−</mo><mn>2</mn><mi>N</mi></mrow></msub></math> topological orders enriched by a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>6</mn><mi>m</mi><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo><mo>×</mo><msubsup><mi mathvariant=\"double-struck\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>4</mn><mi>m</mi><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo><mo>×</mo><msubsup><mi mathvariant=\"double-struck\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>6</mn><mi>m</mi><mo>×</mo><msubsup><mi mathvariant=\"double-struck\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math>, or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>4</mn><mi>m</mi><mo>×</mo><msubsup><mi mathvariant=\"double-struck\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math> symmetry, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>6</mn></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>4</mn></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>6</mn><mi>m</mi></math>, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mn>4</mn><mi>m</mi></math> are lattice symmetries while SO(3) and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mi mathvariant=\"double-struck\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math> are spin rotation and time-reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mi mathvariant=\"double-struck\">Z</mi><mn>2</mn></msub></math> topological order.","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":null,"pages":null},"PeriodicalIF":11.6000,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Classification of Symmetry-Enriched Topological Quantum Spin Liquids\",\"authors\":\"Weicheng Ye, Liujun Zou\",\"doi\":\"10.1103/physrevx.14.021053\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary symmetry, and may permute anyons. The framework applies to all types of lattices and can systematically distinguish different lattice systems with the same symmetry group using their quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi mathvariant=\\\"normal\\\">U</mi><mo stretchy=\\\"false\\\">(</mo><mn>1</mn><msub><mo stretchy=\\\"false\\\">)</mo><mrow><mn>2</mn><mi>N</mi></mrow></msub></math> chiral states and non-Abelian <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mrow><msup><mrow><mi>Ising</mi></mrow><mrow><mo stretchy=\\\"false\\\">(</mo><mi>ν</mi><mo stretchy=\\\"false\\\">)</mo></mrow></msup></mrow></math> states enriched by a <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>6</mn><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\\\"false\\\">(</mo><mn>3</mn><mo stretchy=\\\"false\\\">)</mo></math> or <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>4</mn><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\\\"false\\\">(</mo><mn>3</mn><mo stretchy=\\\"false\\\">)</mo></math> symmetry and <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msub><mi mathvariant=\\\"double-struck\\\">Z</mi><mi>N</mi></msub></math> topological orders and <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi mathvariant=\\\"normal\\\">U</mi><mo stretchy=\\\"false\\\">(</mo><mn>1</mn><msub><mo stretchy=\\\"false\\\">)</mo><mrow><mn>2</mn><mi>N</mi></mrow></msub><mo>×</mo><mi mathvariant=\\\"normal\\\">U</mi><mo stretchy=\\\"false\\\">(</mo><mn>1</mn><msub><mo stretchy=\\\"false\\\">)</mo><mrow><mo>−</mo><mn>2</mn><mi>N</mi></mrow></msub></math> topological orders enriched by a <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>6</mn><mi>m</mi><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\\\"false\\\">(</mo><mn>3</mn><mo stretchy=\\\"false\\\">)</mo><mo>×</mo><msubsup><mi mathvariant=\\\"double-struck\\\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math>, <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>4</mn><mi>m</mi><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy=\\\"false\\\">(</mo><mn>3</mn><mo stretchy=\\\"false\\\">)</mo><mo>×</mo><msubsup><mi mathvariant=\\\"double-struck\\\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math>, <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>6</mn><mi>m</mi><mo>×</mo><msubsup><mi mathvariant=\\\"double-struck\\\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math>, or <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>4</mn><mi>m</mi><mo>×</mo><msubsup><mi mathvariant=\\\"double-struck\\\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math> symmetry, where <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>6</mn></math>, <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>4</mn></math>, <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>6</mn><mi>m</mi></math>, and <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi><mn>4</mn><mi>m</mi></math> are lattice symmetries while SO(3) and <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msubsup><mi mathvariant=\\\"double-struck\\\">Z</mi><mn>2</mn><mi>T</mi></msubsup></math> are spin rotation and time-reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msub><mi mathvariant=\\\"double-struck\\\">Z</mi><mn>2</mn></msub></math> topological order.\",\"PeriodicalId\":20161,\"journal\":{\"name\":\"Physical Review X\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":11.6000,\"publicationDate\":\"2024-06-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Physical Review X\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1103/physrevx.14.021053\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review X","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevx.14.021053","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Classification of Symmetry-Enriched Topological Quantum Spin Liquids
We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary symmetry, and may permute anyons. The framework applies to all types of lattices and can systematically distinguish different lattice systems with the same symmetry group using their quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify chiral states and non-Abelian states enriched by a or symmetry and topological orders and topological orders enriched by a , , , or symmetry, where , , , and are lattice symmetries while SO(3) and are spin rotation and time-reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar topological order.
期刊介绍:
Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.