Supersymmetry and multicriticality in a ladder of constrained fermions

1-systematic study of the phases in a spin-less fermion model with a supersymmetric multi-critical point
2-clear phenomenological picture of the ground and excited states in limiting cases
3-criticalities in floating phases

1-The explanation for the algorithm in Appendix A can be improved.

The authors study the phase diagram of a constrained spinless fermion model on a zigzag ladder. The model possesses a supersymmetric point at which the ground state degeneracy is extensive. The perturbation away from the supersymmetric point lifts this massive degeneracy and leads to either gapless or gapped phase, depending on the relative strength of the third and fourth neighbor interactions. An extensive DMRG study reveals that (i) the gapless phase is an incommensurate Luttinger liquid phase described by c=1 CFT and (ii) the transition between the gapless and gapped phases is in the Pokrovsky-Talapov universality class. In addition, in some limiting cases, the authors give clear phenomenological pictures of the ground states and excited states, which are consistent with what they found numerically. I think the results are interesting, and the paper is written very nicely. Thus, I recommend the manuscript for publication after the authors consider the following comments and suggestions.

- PXP model
In the first paragraph of the Introduction, the authors classified the PXP model as a model with a kinetically constrained hopping. However, the Hamiltonian of the PXP model consists solely of the constrained local spin-flip terms and does not have any kinetic term. Thus, I think the corresponding sentence should be rephrased with more careful wording.

- Eq. (7)
It is not easy to understand what this equation means unless one writes down the 1-particle Hamiltonian. As far as I understand, N=5 is special in that the 1-particle Hamiltonian is written as a 5x5 matrix with all matrix elements 1. I suggest that the authors write down the explicit Hamiltonian just like Eq. (9).

- Eq. (9)
There must be a typo in this equation. I think the (3,3) element of this matrix should be 1. Otherwise, {\bm v}_{\rm GS} below Eq. (9) cannot be a zero-energy state.

- Criticality at the supersymmetric point
I wonder if the authors can study the nature of the criticality right at the supersymmetric point. I know the massive ground state degeneracy at the point makes the standard stuff like the central charge meaningless. Nevertheless, I still wonder if one can talk about the criticality of the ground state sector by sector. Are there any particular fermion densities N_f/N at which the low-energy states are not so degenerate, and the standard CFT machinery applies? (This question might be related to the second comment raised by the other referee: are there any chemical potentials for which the standard analysis makes sense?)

- Boundary of the period-3 phase
I like the phenomenological argument presented in Sec. 3.3, as it is intuitive and easy to understand. However, I do not quite see how the phase boundaries obtained by that argument are consistent with the actual boundaries obtained numerically. In the current manuscript, the authors just state that they agree with each other. I wonder if the authors can show an enlargement of the phase diagram around the phase boundary of the period-3 phase and show the comparison between numerical and phenomenological results, which would be helpful.

- Uniqueness of the ground state(s)
The authors found that the states Eq. (18) are (19) ground states of the model along the line V_3 = V_4 in the period-4 phase. I am curious about whether they are the unique ground states or there are some other ground states. Can the authors prove that there are no other ground states by using the Perron-Frobenius theorem etc.?

- Appendix A
This appendix is just hard to understand. I have almost no idea about what Figures 12 (a)-(e) mean. What does the fusion graph mean? Does the MPO here mean the MPO expression for the Hamiltonian? I would like to suggest that the authors elaborate on them in both the main text and the caption of Fig. 12.

1. First paragraph of Sec. 3.1:
know in the literature ... -> known in the literature ...

The authors take a starting point a fermion ladder with extensive ground-state degeneracies following from supersymmetry. They perturb the couplings, breaking the supersymmetry and most of the degeneracies, and analyze this model in great depth. Much of the phase diagram is a c=1 incommensuate phase, but the supersymmetric point is a multicritical point separating nearby period-4 and period-5 ordered phases as one might have guessed. They also find a period-3 phase where the density is at its maximum allowed value.

I think this paper is a solid piece of work, coming from an impressively thorough numerical analysis. Although there are no great surprises, it is well worth knowing how these peculiar supersymmetric points fit into a larger picture. I thus support its publication in SciPost.

I have a few comments.

-- I wouldn't say (as the authors do in the abstract) that they have "resolved the issue" of extended degeneracies in the supersymmetric models. They've certainly thoroughly understood one particular case, but I'd say the fully 2d cases are still rather mysterious. I expect the answers there to be much subtler.

-- Early on, they fix the chemical potential to be U=-1 (the supersymmetric value). I appreciated the need to make the problem tractable, but it would nice to know that nothing particularly depends on this choice. So whereas I wouldn't demand any major work, it would be nice to be reassured via a few checks that nothing dramatically changes if this fixing is relaxed a bit.

-- The authors state that in the period-4 phase there is an exact ground state along a line segment. They give the exact expression for filling exactly 1/4 (and defined a modified model governing the case where the number of sites is even but not a multiple of 4), but don't display anything specific for other cases. Is the ground state really exact in all cases, or just very close? If the former, they should explain why they haven't given the exact ground state, and if the latter, they should explain. I would expect the former, but would just like to know. Also, presumably if $U\ne -1$, they're no longer exact?

--Re the second paragraph of the conclusion: while some speculation is allowed, I worry somewhat that if the hard-core constraint is relaxed along with the supersymmetry, the multicriticality will collapse. Doesn't Landau theory suggest that? If it doesn't, it would be really interesting to understand why not. Probably the authors should clarify this (or correct me if I've misunderstood).

-- Re the third paragraph of the conclusion: At least as I understand it, the results of Trebst et al are protected by the "topological symmetry" imposed. So whereas another look at these results would be an excellent idea, I wouldn't expect them to change without breaking the symmetry.

-- Re the fourth paragraph of the conclusion: by "problems" what do they mean?? Extensive ground-state degeneracy and what else?