A free fermions in disguise model with claws

The authors find a new model which exhibits "free fermions in disguise" (ffd) behavior, generalizing in a non-trivial and interesting way the original 2019 construction. A sufficient condition for the ffd construction to work is that the "frustration graph" of the Hamiltonian obey certain conditions. (Each vertex of the frustration graph is a term in the Hamiltonian that squares to 1; each edge corresponds to two terms that anticommute with each other.) A few years ago, Balazs Pozgay (one of the authors here) and I subsequently found a model (henceforth the FP model) that violated these graphical conditions but which still could be solved using a generalization of the original construction.

The model in this paper by Fukai, Vona and Pozsgay likewise violates these graphical conditions. The authors clearly explain how to solve it, generalizing these methods again. This model is in a sense the simplest example of a ffd model violating the graphical conditions. This model requires adding no new degrees of freedom to the ffd model, so the Hamiltonian can be viewed as a deformation of the original one. The FP model, on the other hand, is in essence two copies of the original coupled in a specific way. Moreover, the authors provide a more general method, giving hope that the graph-theory construction can be extended to such ffd models.

For these reasons I believe the paper deserves publication in SciPost. It solves a novel model using a novel construction.

The authors do not, however, note the considerable similarities with the FP model and its solution. Indeed, the authors state that they believe theirs is the first ffd model with "even holes" in the frustration graph. That's not true; the FP model has them. Moreover, their construction relies on noticing that the operators comprising an even hole implements a gauge symmetry, and its effect can be removed by fixing the gauge. The FP has the same structure; it's just that the gauge fixing there is built in from the beginning.

This observation of gauge symmetry is interesting and definitely an extension of the work In the FP paper. It would be useful, however, to make this connection clearer.

1) As indicated in the report, the authors need to explain better the connection with the FP model, and remove the claim that this is the first ffd example with even holes.

2) It makes sense to first introduce the model and its frustration graph before embarking on the proof of its ffd behavior. The authors indeed do that in section 3. However, they should reword a little to make it clearer that they are summarizing the results here, and indicate better where in the paper they prove it. For example toward the end of section 3.4, they state that the graphs in Fig 8 are free fermion, but don't say how they know that. In the caption of Fig 8 they state that these graphs "can be proven" to be free fermionic, but again don't indicate where. Likewise, the relation 3.11 is rather mysterious as it stands; an indication of where and how it gets used would be useful.

3) In the paragraph beginning "Among the remaining couplings..." in sec 3.4, don't they mean equation 3.13, not 3.11? (The former follows from the latter.)

4) In the following paragraph, they say that finding currents makes a model free fermionic, but that's not quite right -- the currents (presumably) make the model integrable, but not necessarily free fermionic. An example is the original ffd model, which has such currents for periodic b.c., but is not free-fermionic.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1- Clarity: this paper is very clear. The problem is clearly defined, the proofd are well organised. The notation is clear. The derivations are thoughtfully expressed.

2- Novelty: while this result (the existence of such a model, with a graphical recognition and decomposition) is not surprising, the simplicity of the expression in this model is surprising.

I have no weaknesses of this paper.

This is a fantastic paper. I believe that it clearly meets the requirements of correctness, impact, novelty and clarity of expression. This paper is an extension of a very beautiful direction of research linking classical graph theory with the expansion of exact solvability, and it has made a clear and impressive step beyond the previously known.

I believe I noticed a few spelling errors and turns of phrase that were slightly confusing but I do not request any substantive changes

Publish (easily meets expectations and criteria for this Journal; among top 50%)