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 <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