Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry

By Liang Kong, T. Lan, X. Wen, Zhi-Hao Zhang, Hao Zheng. 2020

195 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/21564f2ebdf2aa6d2cf59c3aa4c3700f6ef321a0

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In Atomic and Molecular Physics, and Optics

In Nuclear and High Energy Physics

Citations in the corpus

Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond (2022, background): “They have also been realized on the lattice [119–125,108,126–128], sometimes under the name of “algebraic higher symmetry” [129, 45].”
Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions (2022, background): “This is known as the higher condensation [52, 66, 54, 55, 65] or higher gauging [19].”
Higher Gauging and Non-invertible Condensation Defects (2022, background): “Motivated by the discussion in [54,55, 32], the resulting topological defect will be called a condensation defect.”