Lattice Models for Phases and Transitions with Non-Invertible Symmetries

The authors study a lattice-model realization of (1+1)-dimensional gapped phases with generic catogorical (also known as non-invertible) symmetries, as well as second-order phase transitions connecting them. The lattice models are constructed through anyonic chains. The paper is interesting because it provides a general construction of phases and phase transitions, through which general properties of those phases can be studied. Many examples are provided. The paper also lays the ground towards higher-dimensional generalizations. I definitely recommend the publication in SciPost Physics.

I ask some minor modifications with respect to the v4 available on arXiv. In my opinion, the introduction is not very clear and should be improved, given the high quality of the paper, to the benefit of the reader. In particular, the following points should be clarified.

* From both the abstract and the introduction, it is not very clear whether the authors are working in generic dimension $d$ or only in (1+1) dimensions. Since the latter is the case, this should be more clearly stated both in the abstract and in the introduction.

* The paragraph "Anyon-chains from SymTFT data," and partly the following paragraph "Twisted sectors," are not very clear, in particular what is the configuration of boundaries, object $\rho$, and lines responsible for twisted sectors. Probably a picture at this point would be very helpful.

* I noticed that all examples considered in the paper are of Abelian symmetries. This is surely already a very rich class of examples, due to non-invertibility, however I think that this fact should be mentioned in the introduction. On the other hand, what about nonAbelian examples? Could they also be treated with the authors' methods and tools? A comment on this would be helpful.

* Finally, as described in Section 2.1, the models studied in this paper are defined on the circle (with periodic or twisted boundary conditions). This should be more clearly mentioned in the introduction. In some later work they will consider open chains with two boundary conditions: this could be mentioned in the paragraph "Outlook" at page 6.

Ask for minor revision

1. The paper studies anyonic chains with group-like and non-invertible symmetries in a systematic way. The construction of the models (Hilbert space, twisted Hilbert space, action of the symmetry, order parameters) is carefully reviewed.
2. The paper shows the connection with the SymTFT – an important tool in field theory, that allows for the classification of phases and phases transitions. This is an important unification between the high-energy literature and the condensed matter literature.
3. The paper shows how the SymTFT contains the information needed to construct a lattice model for gapped and gapless phases with a given symmetry.
4. The paper explains many background aspects that are needed to understand the paper. Even though sometimes this is not enough for a non-specialist to understand the content, as the subject is highly technical, the effort is highly appreciated.
5. The paper shows both the general construction and explicit detailed examples.

1. The paper is sometimes difficult to understand.

The paper is of excellent quality. It provides a link between the condensed matter and high-energy communities by studying lattice models (common in the condensed matter community) from the SymTFT perspective (developed in the high-energy community). It is well written and includes multiple detailed examples.

The acceptance criteria are met. I recommend publication, provided that a few questions are addressed and some minor changes are made.

1. Page 4, paragraph 'anyon-chains from SymTFT data'. Mention that the boundaries are boundaries of the SymTFT, and that they are 2d (at this point of the paper, one could still think that you are talking about 1d boundaries of the physical theory. Mention that the chain is closed.

2. Are the 'generalized charges' irreps of Tube(S), where S is the symmetry category? If yes, this could be written explicitly.

3. Explain or provide a reference for the fact that the lines in Z(S) that label the generalized charges need to be bosonic. In the literature (e.g. arXiv:2208.05495) it is common to find Rep(Tube(S)) = Z(S), with no comment on the bosonic nature.

4. Page 7, give a reference for the sentence "but this is easily remedied by choosing specific basis of the Hilbert space (and representation of the Hamiltonian on this basis in terms of standard Heisenberg matrices)."

5. Around equation (2.4), please refer to a simple example (like the ones done later in the text) in which the coefficients of (2.4) are computed. Same for equation (2.18)

6. Question: In figure 1, the lines of type $\rho$ do not end on any operator, right? So, shouldn't they be contractible when put on the topological boundary? Wouldn't this make the lattice disappear? In other words, is the lattice model really embedded on the boundary of the SymTFT or is it more like a metaphor?

7. Above (2.16), ref [18] is cited twice

8. Above (2.51), in the sentence "Being dual to the gauged Zb2 symmetry, the end-point of P, i.e., vP is charged under the ends of the b lines from above": is it correct to say that the endpoint of a line is dual to some symmetry? Maybe the sentence can be clarified a bit

9. How do you go from the second to the third line of (2.70)? If you also think this is not trivial, you could add a comment.

10. Could you give a reference or give some explanation to help the reader deriving the lagrangian algebras in (2.77)? For instance, I thought that the Lagrangian algebra associated to VecG was of the form $\oplus_r \operatorname{dim} r \, ([1], r)$ with $r$ an irrep of $G$, like in arXiv:2305.1715 eq (4.161)

11. Between (2.77) and (2.78): Could you add some information about what functor is used to project the bulk lines to the boundary lines? It seems that it is not just the forgetful functor.

12. In (2.80): what is $\vec g, g$?

13. If the choices of $m$ and $\Delta$ appearing in (2.89) were already studied in the literature, it would be helpful to give a reference. Is $\Delta$ really a coproduct in some sense?

14. In paragraph 3.2.1, it is declared what is $\mathcal C, \mathcal M$ and so $\mathcal S$. However, $\rho$ is not declared. Is it that an oversight? If not, you could comment

15. Curiosity: is there some specific reason for which in (3.23) you choose $\mathfrak B^{\mathrm{sym}} = 1 \oplus e$ and not $1 \oplus m$?I thought that $1 \oplus m$ was more natural as it makes the line with rep label condense, and the line with group label can stay on the boundary. I also agree that they both produce a boundary with VecZ2 symmetry, I am just wandering if it is just an arbitrary choice of if I am missing something.

16. In (3.24), you declare the available Frobenius algebras. Could you provide information about the product and the coproduct? Also, now that $\rho (=A)$ is declared, it would be helpful to discuss briefly the Hilbert space. For $A=1$, it is exactly two dimensional, isn't it?

17. In 3.2.2, it would be helpful to repeat that C = M = VecG

18. In (3.40), what is the $\prod_j$ doing? Shouldn't you just have $\sum_{h_1, \dots h_L}$? The same product also appears in (A.21), and maybe somewhere else.

19. Below (3.46), in the sentence "The defining property of SPT ground states is that their ground state acquires a non-trivial charge" there is something strange.

20. I was not really able to understand the general construction of models for phase transitions, and this is probably my fault. However, it would be helpful if, in the first example of the Z4 SSB to Z2 SSB transition, there were a short summary of the initial data, the goal, the main steps to be performed, and eventually the arbitrary choices that can be made.

Ask for minor revision

In this paper, the authors study lattice models with various non-invertible symmetries and study their phase diagrams. The subject is at the forefront of the current research in theoretical physics, and the obtained results are quite interesting. I recommend publication of this paper.
I just have one request: can the authors summarize/showcase the lattice models (and maybe phase diagram -- although it would be also fine to give pointers to the relevant sections) they constructed, in a single place (either in Introduction, Summary, or even Appendix)? It requires a lot of effort and time to go through the paper especially for non-experts on the subject. However, even non-experts could check the lattice models and play with them numerically if presented accessibly.

Ask for minor revision