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Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions
Name
PhysRevResearch.2.033417.pdf
Description
Published version
Size
1.02 MB
Format
Adobe PDF
Checksum (MD5)
d280aafb8e590de66823c64fc4fda920
Author(s)
Ji, Wenjie
Wen, Xiao-Gang
Date Issued
2020
Journal
Physical Review Research
Publisher
American Physical Society (APS)
Citation
Ji, Wenjie and Wen, Xiao-Gang. 2020. "Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions." Physical Review Research, 2 (3).
Version
Final published version
Abstract
For a zero-temperature Landau symmetry breaking transition in $n$-dimensional
space that completely breaks a finite symmetry $G$, the critical point at the
transition has the symmetry $G$. In this paper, we show that the critical point
also has a dual symmetry - a $(n-1)$-symmetry described by a higher group when
$G$ is Abelian or an algebraic $(n-1)$-symmetry beyond higher group when $G$ is
non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of
$G$-gauge theory in one higher dimension. The conservation of gauge charge and
gauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dual
symmetry respectively. So any $G$-symmetric system actually has a larger
symmetry called categorical symmetry, which is a combination of the symmetry
and the dual symmetry. However, part (and only part) of the categorical
symmetry must be spontaneously broken in any gapped phase of the system, but
there exists a gapless state where the categorical symmetry is not
spontaneously broken. Such a gapless state corresponds to the usual critical
point of Landau symmetry breaking transition. The above results remain valid
even if we expand the notion of symmetry to include higher symmetries and
algebraic higher symmetries. Thus our result also applies to critical points
for transitions between topological phases of matter. In particular, we show
that there can be several critical points for the transition from the 3+1D
$Z_2$ gauge theory to a trivial phase. The critical point from Higgs
condensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and its
dual - a $Z_2$ 2-symmetry, while the critical point of the confinement
transition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual
- another $Z_2$ 1-symmetry.
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Creative Commons Attribution 4.0 International License
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DOI of Published Version
10.1103/PHYSREVRESEARCH.2.033417
