Categories of quantum liquids I
Nadine Wagener,Nadine Wagener +1 more
TL;DR: In this article , a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion was developed, which includes topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases.
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Abstract: We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion. Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their higher centers. We also outline a theory of unitary higher categories based on a $*$-version of condensation completion. After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all $n$D quantum liquids form a $*$-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.
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References
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One dimensional gapped quantum phases and enriched fusion categories
TL;DR: In this article, the authors used Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases.
4
A factorization homology primer
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On the classification of topological orders
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