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Exotic Invertible Phases with Higher-Group Symmetries

by Po-Shen Hsin, Wenjie Ji, Chao-Ming Jian

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 12, 052 (2022)

Submission summary

As Contributors: Po-Shen Hsin · Chao-Ming Jian
Preprint link: scipost_202109_00022v2
Date submitted: 2021-11-29 00:41
Submitted by: Hsin, Po-Shen
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We investigate a family of invertible phases of matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev's chain in 1+1d. The excitation has $\mathbb{Z}_2$ higher-form symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry. We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the $\mathbb{Z}_2$ one-form symmetry and the time-reversal symmetry, and has surface thermal Hall conductance not realized in conventional time-reversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the $SO(3)_-$ gauge theory with unit discrete theta parameter, which enjoys the same spacetime two-group symmetry. We discuss several applications including the analogue of ``fermionization'' for ordinary bosonic theories with $\mathbb{Z}_2$ non-anomalous internal higher-form symmetry and time-reversal symmetry.

Current status:
Has been resubmitted

Author comments upon resubmission

We thank the referees and editors for their time and for thoroughly reviewing and improving our work.

List of changes

- The grammars and spellings mentioned in the reports are fixed, as well as the references (repeated DOI, capitalized titles, and math symbols).
- p3 added clarification in the second paragraph that the 1+1d theory is the non-trivial phase of the Kitaev's chain.
- p3 added clarification in footnote 2 about the terminology of unfaithful higher form symmetry i.e. symmetry generator invariant under small deformations of the submanifold where the generator is supported.
- p7 added (2.7) and an explanation that the theory (2.6) is invertible i.e. gapped with a unique ground state. The explanation is referred to later in the paragraph below (3.21) and (4.4).
- p16: below (3.22) correct the non-causal reference ``will be discussed in Section 3.1" -> "as we discussed in Section 3.1".
- p19: below (3.32) added clarification about the Pontraygin square P and the quadratic function q.
- p25: added footnote 25 using anti-semion as an example to explain the chiral central charge mentioned here.
- p29 beginning of Section 3.6, added clarification that the SO(3)- theory discussed here has the discrete theta angle p=1 (as opposite to p=3).
- p29 beginning of Section 3.6, added clarification that "m=3" stands for the Z_2 one-form symmetry SPT phase with the partition function (E.1) with m=3.
- p33: in equation (3.66) added clarification about where b cup b comes from (difference of q(b) and -q(b)).
- p34 figure 3 caption: added that the analogous 1+1d action for (3.68) is given by Z2 gauge theory+ Ising scalar as in (2.9) of Ref [55], and it is dual to free massless Majorana fermion, with the fermion mass identified with the mass square of the Ising scalar.
- p35: in (3.68) changes the sign of lambda_{12}.
- p35: in the bullet point M^2<0, lambda_{12} is replaced by lambda'.
- p49: in Appendix B added a final paragraph about a construction of the quadratic function using the Wu3 structure.
- p50 footnote 49: added that the general SL(2,Z) map is not a diffeomorphism, while the mapping class group is D8.

Reports on this Submission

Anonymous Report 3 on 2021-12-4 (Invited Report)


The authors have addressed satisfactorily most of the questions. I recommend the paper for publication in the present form.

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Anonymous Report 2 on 2021-11-30 (Invited Report)


The revision is mostly satisfactory except for the addition to Appendix B.

There should be a mathematical theorem saying that a choice of the quadratic refinement is in 1-to-1 correspondence with the choice of the trivialization of the Wu structure. (For the simplest case of the Wu structure, i.e. the spin structure, this was done by Atiyah using index theorem and then by Johnson more elementarily.)

As this is a physics paper, the authors do not have to explain it, but they at least have to provide a reference.

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Author:  Po-Shen Hsin  on 2021-12-01  [id 1997]

(in reply to Report 2 on 2021-11-30)

Thanks for the comment. The correspondence is explained in Ref. [35] (eg. Corollary 1.17).

Anonymous on 2021-12-27  [id 2056]

(in reply to Po-Shen Hsin on 2021-12-01 [id 1997])

I'm the referee and the authors were quite right, the point I raised was already in [35] which was already properly cited in v2 from Appendix B. I am thankful to the authors (and I am sorry for making them going through the trouble) to provide the new version v3 with an additional sentence in Appendix B to emphasize the correspondence between Wu structure and the quadratic function. I am also very sorry that I did not notice the authors' comment earlier and that my reply was very slow.

I think the v3 can be published as is.

Anonymous Report 1 on 2021-11-29 (Invited Report)


The manuscript is now ready for publication.

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