Dualities in One-Dimensional Quantum Lattice Models: Topological Sectors
Laurens Lootens, Clement Delcamp, Frank Verstraete
Abstract
It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mfrac><a:mn>1</a:mn><a:mn>2</a:mn></a:mfrac></a:math> Heisenberg <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi>X</d:mi><d:mi>X</d:mi><d:mi>Z</d:mi></d:math> model. One model has symmetry operators forming the fusion category <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mrow><g:mi mathvariant="sans-serif">R</g:mi><g:mi mathvariant="sans-serif">e</g:mi><g:mi mathvariant="sans-serif">p</g:mi></g:mrow><g:mo stretchy="false">(</g:mo><g:msub><g:mrow><g:mi mathvariant="script">S</g:mi></g:mrow><g:mn>3</g:mn></g:msub><g:mo stretchy="false">)</g:mo></g:math> of representations of the group <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><p:msub><p:mrow><p:mi mathvariant="script">S</p:mi></p:mrow><p:mn>3</p:mn></p:msub></p:math>. We find that the mapping between its topological sectors and those of the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><t:mi>X</t:mi><t:mi>X</t:mi><t:mi>Z</t:mi></t:math> model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><w:mrow><w:mi mathvariant="sans-serif">R</w:mi><w:mi mathvariant="sans-serif">e</w:mi><w:mi mathvariant="sans-serif">p</w:mi></w:mrow><w:mo stretchy="false">(</w:mo><w:msub><w:mrow><w:mi mathvariant="script">S</w:mi></w:mrow><w:mn>3</w:mn></w:msub><w:mo stretchy="false">)</w:mo></w:math>. Published by the American Physical Society 2024