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Subsystem Non-Invertible Symmetry Operators and Defects

by Weiguang Cao, Linhao Li, Masahito Yamazaki, Yunqin Zheng

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Submission summary

Authors (as registered SciPost users): Weiguang Cao
Submission information
Preprint Link: scipost_202308_00011v1  (pdf)
Date accepted: 2023-09-08
Date submitted: 2023-08-08 09:10
Submitted by: Cao, Weiguang
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We explore non-invertible symmetries in two-dimensional lattice models with subsystem $\mathbb Z_2$ symmetry. We introduce a subsystem $\mathbb Z_2$-gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.

List of changes

1. We fix all typos the referee mentioned.

2. We add a comment in the end of section 1.2
"Here, the subsystem KW duality symmetry has co-dimension 1 non-invertible symmetry operator and defect, which is different from the co-dimension 2 invertible subsystem $\mathbb Z_2$ symmetry operators and defects. Furthermore, the non-invertible fusion rule will mix operators (defects) of different co-dimensions."

3. We include footnote 5 in the introduction to clarify terminologies.
"We mainly use the terminology "operators" for maps from one Hilbert space (defined on sites) to another Hilbert space (defined on links or plaquettes) and "defects" for interfaces between two theories. One can further redefine the link/plaquette Hilbert spaces to be supported on sites, which we briefly discuss in Sec. 2.4 and 3.4. This redefinition makes the operators to act within on one Hilbert space, and the defects between a single theory. This is consistent with the recent discussion [153]."

4. We include section 2.4 and 3.4 to discuss about operators and defects that exist in a single theory.

Published as SciPost Phys. 15, 155 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-8-21 (Invited Report)


The discussion has become clearer after authors made some revisions. I recommend this paper for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2023-8-16 (Invited Report)


I now recommend the draft for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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