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Gapped Phases with Non-Invertible Symmetries: (1+1)d
by Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer, Sakura Schafer-Nameki
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.