A range three elliptic deformation of the Hubbard model

Please find the attached pdf file.

The submitted manuscript introduces two integrable modifications of the 1-dimensional Hubbard model. One represents a deformation with interactions ranging over three sites, the other consists of standard nearest-neighbour interactions which, however, do not preserve spin. These two models are related by a bond-site transformation, and their underlying R-matrix is derived. It depends on both spectral parameters (individually) through elliptic functions, and it represents a novel solution of the Yang-Baxter equation. These results leading to a new and very non-trivial quantum R-matrix exciting, and they call for further exploration and classification. For these reasons I recommend publication in SciPost.

In the following I list some remarks that the authors should address towards publication:

1) Brackets in equation (2.4) seem to be missing

2) Above equation (2.15), Identity is spelled with a capital I

3) The paragraph above equation (3.4) mentions a new formalism for integrability of medium-range interaction developed in reference [32] and to be reviewed further in paragraph 5.2. It may help the reader to already here sketch some key element(s) of this formulation, i.e. what is this about?

4) equation (5.6) seems to be missing a log as in (5.5) which is also mentioned just below the equation.

5) I don't quite follow the claims around (5.5): It seems that regularity of the Lax operator would ensure a local Hamiltonian, but I do not see how regularity of the R-matrix would help.

6) Equation (5.8) defines the relationship between L and R with an unusual dependency on the two parameters. Does this work generically or specifically in this model and/or with the particular transformation of the parameters

7) Together items 6) and 7) might work, but taking the relationship of parameters into account shouldn't one consider the point u=mu/alpha(mu) for L(u,mu) such that R is evaluated at R(mu, mu)?

8) It is worthwhile to point out that the product/quotient of the two quantities (5.10) is an elliptic functions. Hence, only one of the two escapes the elliptic property. Similarly, it would suffice to use just one of the functions f and g in appendix A.

9) Is the expression in equation (5.12) really correct? For a standard flow of the tensor sites, one that is naturally compatible with equation (5.13), the two permutation operators Paj and Pbj should be exchanged. Then effectively Lchech_abj would map site j to site a while ab are shifted to bj. Equation (5.13) would then be a rather natural YBE in the R/Lchech notation with two pairs of sites and a single site. As it stands, Lchech_abj seems to shift a to j and bj to ab, which is the opposite cyclic permutation for what is needed to make (5.13) work out naturally. Finally, defining the Hamiltonian to be a logarithmic derivative, would (naturally) yield local terms only if the correct permutation is assumed.

10) Also (5.17) appears rather odd. Is this specific to the model, or does it follow generically from the bond-site transformation? If so, how?

11) The following relations all appear a bit unnatural, could be related to items 9) and/or 10). However, in the end the authors obtain a rather natural YBE in equation (5.27) for three pairs of sites. Now I'm not really questioning the validity of (5.12) through (5.27) which I have not checked in terms of expressions, but I'm wondering whether it is a coincidence that they hold, and whether a simpler set of relations (based on the permutation in item 9) would (also) hold. A less technical and more descriptive exposition of the would also avoid such potential misunderstandings.

12) Is equation (5.27) the principal outcome of its section? It would seem more straight-forward to derive it as the compatibility condition for equation (5.13).

13) Below equation (5.22): grammar at "depends" and "in" appears wrong.

14) Beginning of section 6): How to combine these two limits? Are these two different limits, two separate limits to be taken in some (any?) order or one simultaneous limit? If so, how to write u and U as limit of a common parameter?

15) Appendix A promises the computer algebra expression of the R-matrix upon request to the authors. For a modern digital publication, it would seem more natural to supply the expression as ancillary digital material to the publication as it represents relevant scientific data. This could be done easily with a minimal mathematica notebook containing the expressions of appendix A in machine readable format.

See pdf attached