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Loops in 4+1d Topological Phases

by Xie Chen, Arpit Dua, Po-Shen Hsin, Chao-Ming Jian, Wilbur Shirley, Cenke Xu

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Submission summary

Authors (as registered SciPost users): Xie Chen · Arpit Dua · Po-Shen Hsin · Chao-Ming Jian · Cenke Xu
Submission information
Preprint Link: scipost_202210_00089v1  (pdf)
Date accepted: 2023-01-10
Date submitted: 2022-10-30 23:54
Submitted by: Dua, Arpit
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form $\mathbb{Z}_2$ gauge field (the loop only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self `exchange' statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The $SL(2,\mathbb{Z}_2)$ symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the `fractional Maxwell theory' and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the $\mathbb{Z}_2$ gauge group to $\mathbb{Z}_N$.

Published as SciPost Phys. 15, 001 (2023)

Submission & Refereeing History

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Resubmission scipost_202210_00089v1 on 30 October 2022

Reports on this Submission

Anonymous Report 1 on 2022-11-28 (Invited Report)


The authors have significantly improved the presentation of results the paper, better explained how the work connects to existing literature, and adequately addressed my technical concerns.

  • validity: good
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: good

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