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Exotic Invertible Phases with HigherGroup Symmetries
by PoShen Hsin, Wenjie Ji, ChaoMing Jian
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Submission summary
Authors (as registered SciPost users):  PoShen Hsin · ChaoMing Jian 
Submission information  

Preprint Link:  scipost_202109_00022v2 (pdf) 
Date submitted:  20211129 00:41 
Submitted by:  Hsin, PoShen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate a family of invertible phases of matter with higherdimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev's chain in 1+1d. The excitation has $\mathbb{Z}_2$ higherform 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 $\mathbb{Z}_2$ oneform symmetry and the timereversal symmetry, and has surface thermal Hall conductance not realized in conventional timereversal 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 twogroup symmetry. We discuss several applications including the analogue of ``fermionization'' for ordinary bosonic theories with $\mathbb{Z}_2$ nonanomalous internal higherform symmetry and timereversal symmetry.
Author comments upon resubmission
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 nontrivial 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 noncausal 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 antisemion 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 oneform 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.
Current status:
Reports on this Submission
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 1to1 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/s222.2.365 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.
Author: PoShen Hsin on 20211201 [id 1997]
(in reply to Report 2 on 20211130)Thanks for the comment. The correspondence is explained in Ref. [35] (eg. Corollary 1.17).
Anonymous on 20211227 [id 2056]
(in reply to PoShen Hsin on 20211201 [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.