Abstract
In this work, we use 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 CFT-type gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial. We also study the categorical descriptions of the boundaries of these models. In the end, we obtain a classification of and the categorical descriptions of all 1-dimensional (spatial dimension) gapped quantum phases with a bosonic/fermionic finite onsite symmetry.
Highlights
In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and CFT-type gapless quantum phases
We show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial
The idea of gauging the symmetry works and can be generalized to higher dimensions [31], it is unsatisfying because the SPT/SET orders are well-defined before the gauging
Summary
In 2d, a categorical description of bosonic SPT/SET orders with a finite onsite symmetry was first introduced by Barkeshli, Bonderson, Cheng and Wang in [2] based on the idea of gauging the symmetry by introducing 1d symmetry defects. A new description for both bosonic and fermionic 2d SPT/SET orders was introduced in [44, 45] based on the idea of gauging the symmetry but in a different way [49]. The idea of gauging the symmetry is to introduce additional particles to S in a minimal way such that all old and new particles can be detected by double braidings again In mathematical language, it amounts to finding a minimal modular extension of S [45, 53]. This dissatisfaction motivated a new description of SPT/SET orders without gauging the symmetry [31] This description is based on the idea of boundary-bulk relation [36, 37]. Using the fact that the bulk is the center of the boundary [36, 37], we obtain a mathematical description of an anomaly-free nd SPT/SET order, summarized in the following physical theorem [31, theoremph 1.1].
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