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Gapped Phases with Non-Invertible Symmetries: (1+1)d
by Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer, Sakura Schafer-Nameki
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Lakshya Bhardwaj · Lea Bottini |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2310.03784v4 (pdf) |
| Date submitted: | 2024-07-29 11:32 |
| Submitted by: | Bottini, Lea |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. We also provide a mathematical description of symmetric gapped phases as 2-functors from delooping of fusion category characterizing the symmetry to Euler completion of 2-vector spaces.
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Author comments upon resubmission
We list below the changes made according to each referee's suggestions.
Report 1:
We thank the referee for the extremely detailed reading of the paper and all the comments.
We have addressed most of the report's requests when they were clearly stated. Regarding reference
https://doi.org/10.1103/PhysRevB.79.045316 we were not sure where the referee wants us to cite this.
Report 2:
We thank the referee for the detailed report and comments.
Regarding the point 2: We have emphasized that the SPT phases are strictly speaking outside of the original Landau paradigm.
Regarding point 3: that is a very good point and we have made our statements clearer to distinguish this from the G-phases that the referee pointed out.
Regarding point 4: In subsequent papers we discuss reducible boundary conditions for topological orders which are important for gapless phases, and a detailed discussion can be found there 2403.00905. In the present paper we indeed focus on simple boundary conditions. As for eqn (3.9), the multi-fusion category being discussed there is the category formed by lines of a 2d TFT that contains multiple vacua, and *not* the category formed by lines of an irreducible 2d topological boundary condition of a 3d TFT.
Regarding point 5: The irreducibility of physical boundaries is emphasized in bold on page 27.
We have addressed all other questions in the text at the referred locations.
Current status:
Reports on this Submission
Report
The authors mostly improved the manuscript along the lines suggested. While I still think that generally more effort should have been put into acknowledging relevant pre-2020 contributions to the field and also making references more specific, i.e. tie them to the concrete discussion of examples and results that are reviewed instead of citing them in bulk in the introductory section, I am willing to leave it with requesting the inclusion of additional references to the discussion of string order parameters. This was already strongly suggested in my previous report but to assist the authors this is now made more concrete below.
Requested changes
At the beginning of Section 4.2 there seems to be substantial scope for including additional references to the use of string order parameters to describe SPT phases that are protected by invertible symmetries. Historically, and in terms of generality the most relevant works in the context of the present paper are probably
* https://doi.org/10.1103/PhysRevB.3.3918
(disorder variable in the Ising model)
* https://doi.org/10.1007/BF02097239
(Z2xZ2 hidden symmetry breaking - original case underlying the Kennedy-Tasaki transformation)
* https://doi.org/10.1088/0953-8984/4/36/019
(Z2xZ2 hidden symmetry breaking - more general case)
* https://doi.org/10.1103/PhysRevLett.100.167202
(general discussion about the relation between string order and symmetries)
* https://doi.org/10.1103/PhysRevLett.109.050402
(string-like order parameters for SPT phases)
* https://doi.org/10.1103/PhysRevB.86.125441
(generalized string order parameters for SPT phases)
* https://doi.org/10.1103/PhysRevB.88.085114
(hidden symmetry breaking picture for general finite abelian groups)
* https://doi.org/10.1103/PhysRevB.88.125115
(similar but with an emphasis on discussing orbits of phases under Kennedy-Tasaki-like dualities)
Hidden symmetry breaking refers to the characterization of SPT phases in terms of string-order parameters that can be mapped to local order parameters using a non-local duality. Especially, the last two papers include a detailed discussion of the interplay between both local and non-local order parameters in a relatively general setting and how they transform into each other under non-local dualities.
All of this appears highly relevant in the context of the present paper and should, in my opinion, be cited in the intro to section 4.2 and in the appropriate examples (like Z2 or Z2xZ2) to give prospective readers a comprehensive and accurate picture of previous developments.
Concerning https://doi.org/10.1103/PhysRevB.79.045316, my impression is that this is one of the first papers discussing the condensation of anyons on the boundary of a topological phase, predating the discussion in [128] in terms of Lagrangian algebras by 8 years. A place where it could be cited is the beginning of the second paragraph in appendix B.
Recommendation
Ask for minor revision