Classification of Symmetry-Enriched Topological Quantum Spin Liquids
Bibliographic record
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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="normal">U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:msub><a:mo stretchy="false">)</a:mo><a:mrow><a:mn>2</a:mn><a:mi>N</a:mi></a:mrow></a:msub></a:math> chiral states and non-Abelian <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mrow><f:msup><f:mrow><f:mi>Ising</f:mi></f:mrow><f:mrow><f:mo stretchy="false">(</f:mo><f:mi>ν</f:mi><f:mo stretchy="false">)</f:mo></f:mrow></f:msup></f:mrow></f:math> states enriched by a <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>p</j:mi><j:mn>6</j:mn><j:mo>×</j:mo><j:mrow><j:mi>SO</j:mi></j:mrow><j:mo stretchy="false">(</j:mo><j:mn>3</j:mn><j:mo stretchy="false">)</j:mo></j:math> or <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>p</n:mi><n:mn>4</n:mn><n:mo>×</n:mo><n:mrow><n:mi>SO</n:mi></n:mrow><n:mo stretchy="false">(</n:mo><n:mn>3</n:mn><n:mo stretchy="false">)</n:mo></n:math> symmetry and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi mathvariant="double-struck">Z</r:mi><r:mi>N</r:mi></r:msub></r:math> topological orders and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi mathvariant="normal">U</u:mi><u:mo stretchy="false">(</u:mo><u:mn>1</u:mn><u:msub><u:mo stretchy="false">)</u:mo><u:mrow><u:mn>2</u:mn><u:mi>N</u:mi></u:mrow></u:msub><u:mo>×</u:mo><u:mi mathvariant="normal">U</u:mi><u:mo stretchy="false">(</u:mo><u:mn>1</u:mn><u:msub><u:mo stretchy="false">)</u:mo><u:mrow><u:mo>−</u:mo><u:mn>2</u:mn><u:mi>N</u:mi></u:mrow></u:msub></u:math> topological orders enriched by a <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>p</cb:mi><cb:mn>6</cb:mn><cb:mi>m</cb:mi><cb:mo>×</cb:mo><cb:mrow><cb:mi>SO</cb:mi></cb:mrow><cb:mo stretchy="false">(</cb:mo><cb:mn>3</cb:mn><cb:mo stretchy="false">)</cb:mo><cb:mo>×</cb:mo><cb:msubsup><cb:mi mathvariant="double-struck">Z</cb:mi><cb:mn>2</cb:mn><cb:mi>T</cb:mi></cb:msubsup></cb:math>, <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi>p</hb:mi><hb:mn>4</hb:mn><hb:mi>m</hb:mi><hb:mo>×</hb:mo><hb:mrow><hb:mi>SO</hb:mi></hb:mrow><hb:mo stretchy="false">(</hb:mo><hb:mn>3</hb:mn><hb:mo stretchy="false">)</hb:mo><hb:mo>×</hb:mo><hb:msubsup><hb:mi mathvariant="double-struck">Z</hb:mi><hb:mn>2</hb:mn><hb:mi>T</hb:mi></hb:msubsup></hb:math>, <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>p</mb:mi><mb:mn>6</mb:mn><mb:mi>m</mb:mi><mb:mo>×</mb:mo><mb:msubsup><mb:mi mathvariant="double-struck">Z</mb:mi><mb:mn>2</mb:mn><mb:mi>T</mb:mi></mb:msubsup></mb:math>, or <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi>p</pb:mi><pb:mn>4</pb:mn><pb:mi>m</pb:mi><pb:mo>×</pb:mo><pb:msubsup><pb:mi mathvariant="double-struck">Z</pb:mi><pb:mn>2</pb:mn><pb:mi>T</pb:mi></pb:msubsup></pb:math> symmetry, where <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mi>p</sb:mi><sb:mn>6</sb:mn></sb:math>, <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>p</ub:mi><ub:mn>4</ub:mn></ub:math>, <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi>p</wb:mi><wb:mn>6</wb:mn><wb:mi>m</wb:mi></wb:math>, and <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mi>p</yb:mi><yb:mn>4</yb:mn><yb:mi>m</yb:mi></yb:math> are lattice symmetries while SO(3) and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msubsup><ac:mi mathvariant="double-struck">Z</ac:mi><ac:mn>2</ac:mn><ac:mi>T</ac:mi></ac:msubsup></ac: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 <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"><dc:msub><dc:mi mathvariant="double-struck">Z</dc:mi><dc:mn>2</dc:mn></dc:msub></dc:math> topological order. Published by the American Physical Society 2024
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".