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The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
by Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer, Sakura Schafer-Nameki
Submission summary
| Authors (as registered SciPost users): | Lea Bottini · Sakura Schäfer-Nameki |
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| Preprint Link: | scipost_202503_00011v1 (pdf) |
| Date submitted: | 2025-03-07 18:00 |
| Submitted by: | Bottini, Lea |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We provide a generalization of the Symmetry Topological Field Theory (SymTFT) framework to characterize phase transitions and gapless phases with categorical symmetries. The central tool is the club sandwich, which extends the SymTFT setup to include an interface between two topological orders: there is a symmetry boundary, which is gapped, and a physical boundary that may be gapless, but in addition, there is also a gapped interface in the middle. The club sandwich generalizes so-called Kennedy-Tasaki (KT) transformations. Building on the results in [1, 2] on gapped phases with categorical symmetries, we construct gapless theories describing phase transitions with non-invertible symmetries by applying suitable KT transformations on known phase transitions provided by the critical Ising model and the 3-state Potts model. We also describe in detail the order parameters in these gapless theories characterizing the phase transitions, which are generally mixtures of conventional and string-type order parameters mixed together by the action of categorical symmetries. Additionally, removing the physical boundary from the club sandwiches results in club quiches, which characterize all possible gapped boundary phases with (possibly non-invertible) symmetries that can arise on the boundary of a bulk gapped phase. We also provide a mathematical characterization of gapped boundary phases with symmetries as pivotal tensor functors whose targets are pivotal multi-fusion categories.
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Author comments upon resubmission
We thank both referees for their reviews and insightful comments. We reply to reach referee below.
Reply to referee 1:
We thank the referee for their comments. We fully agree that our treatment of condensable algebras is limited as we only consider algebras at the level of objects, without explicitly computing the multiplication morphisms. Let us note that for Lagrangian algebras in Z(C), we know these to be in 1-to-1 correspondence with module categories over C, and for all cases we consider in this work these are classified, so we know our discussion is complete. For condensable algebras, we agree this is a limitation and an important question to explore. In upcoming work, some of the authors (SSN) have computed explicitly the multiplication structures for all the group-theoretical cases and checked that there are no extra algebras. We have added a comment in the introduction below equation I.6 (and in appendix B) stating the limitations of our approach.
Changes: -added clarification around equation I.6 (and in appendix B) as described above; -removed C15 from [138] as necessary condition;
Reply to referee 2:
We thank the referee for their comments. All the points have been addressed in the text. Regarding the comment: "The idea that non-Lagrangian condensation corresponds to gapless boundary is not new. See e.g. arxiv:2205.06244". This is a paper we cite several times, and as we remark in the introduction, this papers studied SymTFT techniques to study gapless phases in the group-like cases. The aim of our paper was precisely to extend these techniques more generally to non-invertible symmetries, where it becomes an essential tool and provides a general framework that can be extended to any higher fusion category symmetry.
Let us add some further clarifications here regarding the requested changes:
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The monoidal functor is more precisely from S to the category of lines living on the boundary B' of the reduced topological order (after colliding the symmetry boundary and the interface). This functor is explicilty discussed in examples, see e.g. IV.53.
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Wording corrected to reflect the fact not any Lagrangian algebra intersection automatically results in a condensable algebra - the maximal common subalgebra must be condensable in that case.
3,5. Is the referee sure about the reference arxiv: 1008.2117? We are aware that our treament of condensable algberas is purely at the level of objects and therefore imprecise. Addressing also the comments of referee 1, we have added a comment in the introduction (below eq. I.6) clearly stating the limitations of this approach.
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In footnote [124], we state 'Here, and in what follows, we assume that we are working with systems having emergent Lorentz symmetry in the infrared.' therfore excluding fractons. We have added bosonic in the text.
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Noted and changed.
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As explained above, we only solve for necessary conditions for condensable algebras, mostly at the level of objects, following e.g. the aforementioned arxiv:2205.06244. Since the conditions listed there are not sufficient, it may happen that an A that satisfies those conditions is not actually a condensable algebra. To determine wheter it is one, we compute the reduced topological order and if this is fully consistent, we assume the algebra is condensable. Equivalently, we reformulate the problem as the computation of a consistent Lagrangian algebras in Z \otimes \bar{Z'}.
We hope we have addressed all the comments of the referees and that the paper can now be recommended for publication.
Current status:
Reports on this Submission
Report
The revised manuscript stated the limitations of solving for condensable algebras at the object level. The authors claimed that the examples considered are all valid. Accepting this, however, there are still further technical details that needs to be explained. For example, given the detailed structure of the condensable algebra, it is sufficient to 'derive' the product of boundary operators; however, in the truncated treatment in this work, it is essentially still guessing for possible solutions satisfying some necessary constraints (those listed under (IV.99)). Potentially there can be multiple solutions for more complicated examples. In short, the boundary operators are rather 'solved' than 'derived'.
It would be better if the authors could go through their manuscript and correct such over-confident statements regarding their methods.
Recommendation
Ask for minor revision