Building 1D lattice models with $G$-graded fusion category

1) Fills in an obvious hole in the literature on non-invertible symmetries, lattice models, and boundaries of topological order.
2) Clearly written with some simple, analytic examples.
3) Presents both analytic and numerical results.

There are some points that I think can be clarified. See comments/questions below.

This paper presents a construction for an interesting class of models describing boundary theories for symmetry-enriched topological orders. These models are amenable to numerical study, and provide a simple generalization of anyon chain models, which have been useful for studying properties of gapless theories with non-invertible symmetries. The quality of the paper is high, but I would like to receive answers to the following comments/questions along with appropriate edits to the paper before recommending it for publication:

1) Very small comment on p 1: "string operators associated with moving abelian anyons in the 2D toric code model" may not be the best way to describe 1-form symmetries, because string operators that move anyons are open strings, which do not commute with the Hamiltonian and do not leave the ground state invariant. Perhaps it is better to use "closed string operators in the 2D toric-code model."

2) The paper draws a connection between the boundaries of SETs and their 1D lattice models. Can the connection be made more concrete? For example, a 2+1D SET is described by a set of data including the anyon permutation by G and an H^2(G,A) class. These pieces of data are important for boundary physics because they determine whether or not the boundary can be G-symmetrically gapped. How are these pieces of data read off from the description of the symmetry in 1+1d?

3) Can you comment more on the choice of $a_i$ and the qualitative effects on the physics of the lattice model? In the Ising example, what would happen if you chose $a_0=\psi$ instead of $a_0=1$?

4) Can you comment on the different choices of $\{w_h^z\}$? It seems that if you set $w_0^z=1,w_{h\neq 0}^z=0$ for all sites then you always a $G$ SSB state? Maybe you can add some comments on why the $H^3(G,U(1))$ class only affects $w_g^0$ in the $\mathbb{Z}_2$ example (does a similar result hold for more general $G$?)

5) Right above eq 38, you mention "We are interested in the cases that the models are gapless, which can be described by conformal field theory (CFT)." How do you know that the gapless regions of the phase diagram can always be described by a CFT? In some points in Fig 3, you certainly don't get CFT behavior because you have quadratic dispersion.

6) Right above Sec 3.7, do you expect that for at least some choices of parameters i.e. for some $\{w_h^z\}$ you get decoupled Fib and $\mathbb{Z}_2$ theories? i.e. something like a decoupled golden chain and a trivial $\mathbb{Z}_2$ paramagnet.

7) To be clear, the F moves below Eq 65 are different from the F symbols of the (graded) fusion category? Because they are F symbols of the G-crossed BTC.

8) Should I think of claim (ii) on line 685 as coming from the fact that even if we get an SSB Hamiltonian, the non-tensor-product structure of the Hilbert space projects out all of the ground states except for one? However, if we reduce to the SPT boundary i.e. we set $\mathcal{C}_0=\{1\}$, then we should get degeneracy because we can get SSB boundary theories and the boundary has a tensory product Hilbert space?

9) Regarding fermionic theories (outlook point 1), https://arxiv.org/pdf/2404.19004 might be of interest. It might also be worth citing https://arxiv.org/abs/2304.01262 and https://link.springer.com/article/10.1007/JHEP10(2023)053 in line 68

10) Regarding outlook point 2, I'm confused about why you need the $x_i$ to come from other module categories. You already seem to get access to different phases just by tuning the $\{w_h^z\}$ variables? And the reason for changing the module category is to access other 1+1D phases with the same graded fusion category symmetry. Does this mean that for a given set of $\{w_h^z\}$ for a fixed set $\{x_i\}$ you only get access to a single gapped phase (and gapless regions)? And in order to access other gapped phases you need to change $\{x_i\}$? For example, for the simple $G=\mathbb{Z}_2$ case with $\mathcal{C}_0=\{1\}$, you get both the trivial symmetric phase ($w_h^z=1$ for all $h,z$) and the SSB phase ($w_0^z=1,w_{h\neq 0}^z=0$). Can you comment more on which phases you expect to access given $\{x_i\}$?

11) Regarding outlook point 3, I'm not sure what you mean by maximal category symmetry here? For example, you write "More generally, one may expect a larger category symmetry $\mathcal{Z}(\mathcal{C}_G)$ in the gapless state of the model." However, $\mathcal{Z}(\mathcal{C}_G)$ is braided while the symmetry of a 1+1D system contains only fusion data? It seems that the maximal category symmetry should be another fusion category, not the center of the fusion category which is a braided tensor category?

12) You don't have to do this, but it would be interesting to compare the anomalous vs non-anomalous phase diagram, to make the claim about "larger likelihood to be gapless" more concrete. Here you only have the phase diagram in Fig 3 for the model with the anomalous Z2 symmetry.

Ask for minor revision

This paper gives a systematic construction of 1D lattice models using graded fusion category degrees of freedom. In particular, the authors take the graded fusion category data, define a constrained Hilbert space out of it, and determine the Hamiltonian that satisfies the graded fusion category symmetry. Several example Hamiltonians are then studied numerically and shown to contain gapless regions in the phase diagram. This is an interesting work, especially given the recent interest in categorical symmetries. Using the protocol given in this paper, one can systematically construct models with categorical symmetry that reflect the edge physics of 2D symmetry enriched phases. The paper is very carefully written and is a nice addition to the literature. I only have one minor comment: This paper https://arxiv.org/abs/2110.12882 seems to be on a related topic, although the result seems to be a subset of that in this paper. Can the authors comment on their relation?