# Exotic Invertible Phases with Higher-Group Symmetries

### 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 Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

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

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".
- 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.

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 052 (2022)

Resubmission scipost_202109_00022v3 on 4 December 2021

Resubmission scipost_202109_00022v2 on 29 November 2021
Submission scipost_202109_00022v1 on 17 September 2021

## Reports on this Submission

### Report

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

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

### 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 http://dx.doi.org/10.24033/asens.1205 using index theorem and then by Johnson http://dx.doi.org/10.1112/jlms/s2-22.2.365 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.

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

### 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.

### Report

The manuscript is now ready for publication.

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