Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry
Name
PhysRevResearch.2.043086.pdf
Description
Published version
Size
3.2 MB
Format
Adobe PDF
Checksum (MD5)
95a4be2e15592d18f8834246b05c9a0b
Author(s)
Kong, Liang
Lan, Tian
Wen, Xiao-Gang
Zhang, Zhi-Hao
Zheng, Hao
Date Issued
2020
Journal
Physical Review Research
Publisher
American Physical Society (APS)
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Physics
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1103/PHYSREVRESEARCH.2.043086
