Symmetry breaking

Explicit symmetry breaking

Quantum mechanics

Theoretical physics

Algebraic number

Pure mathematics

Mathematical physics

Anomaly (physics)

Symmetry group

Global symmetry

Physics

Symmetry operation

Mathematics

Homogeneous space

Spontaneous symmetry breaking

Mathematical analysis

Geometry

Rotational symmetry

Symmetry number

Symmetry (geometry)

Mechanics

Local symmetry

Physical Review Research

A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space. A higher symmetry acts on closed submanifolds (i.e., loops and membranes, etc.), and those transformations form a higher group. In this paper, 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 that, when restricted to its symmetric sub-Hilbert space, an algebraic higher symmetry can be fully characterized by a noninvertible gravitational anomaly (i.e., a topological order in one higher dimension). Thus, we also refer to a noninvertible 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; and (4) the classification of gapped liquid phases for bosonic and/or 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.24 MoreReceived 30 June 2020Revised 16 September 2020Accepted 18 September 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043086Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTopological field theoriesTopological phases of matterTopological quantum computingQuantum InformationParticles & FieldsCondensed Matter & Materials Physics

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