Gapless edges of 2d topological orders and enriched monoidal categories

In this work, we give a precise mathematical description of a fully chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and fully chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk conformal field theories naturally emerge on certain gapless walls between two trivial phases.Comment: 26 pages, 8 figures, An explanation of the appearance of boundary CFT's on a chiral gapless edge, which is based on a generalized "no-go theorem", is added. Final versio

Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also refer to non-invertible gravitational anomaly as categorical symmetry to stress its connection to symmetry. This provides a holographic and entanglement view of symmetries. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low energy excitations, as long as they have the same categorical symmetry; (4) the classification of gapped liquid phases for bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry). This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.Comment: 61 pages, 31 figure

A study of the effect of combination of acupuncture and Chinese medicine (Ban Xia Xie Xin Decoction) on patients with sepsis-induced gastrointestinal dysfunction

Purpose: To investigate the therapeutic effect of combination of acupuncture and Ban Xia Xie Xin Decoction (BXXXD) in patients with sepsis-induced gastrointestinal dysfunction.Methods: A total of 100 patients with sepsis and gastrointestinal dysfunction admitted to our hospitalfrom June 2019 to December 2020 were randomly assigned to a control group and a study group for a retrospective analysis. Both groups received acupuncture treatment, while BXXXD was given to the study group, in addition to acupuncture. Treatment effectiveness, incidence of adverse reactions, degree of recurrence at 1, 2 and 3 months after treatment, gastrointestinal dysfunction scores after 1week of treatment, quality of life index (QLI) scores, frequencies of flatulence and defecation within 48 h, number of bowel rumblings, as well as levels of gastrin and  cholecystokinin (CCK), were determined and compared between the two groups.Results: The group with BXXXD intervention demonstrated superior treatment effectiveness, QLI cores, frequencies of flatulence and defecation within 48 h, number of bowel rumblings, and level of gastrin than those who received only acupuncture treatment (p < 0.05). There was strong evidence of lower incidence of adverse reactions, recurrence at 1, 2, and 3 months after treatment; gastrointestinal dysfunction scores and CCK levels after 1 week of treatment was higher in the study group than in the control group (p < 0.05).Conclusion: The combination of acupuncture and BXXXD is superior to acupuncture alone in sepsis patients with gastrointestinal dysfunction. The combination therapy produces its effect by decreasing clinical symptoms, promoting recovery of the patients, and elevating their quality of life